Modulation and equalization in an orthonormal time-frequency shifting communications system

ABSTRACT

A method for modulating data for transmission within a communication system. The method includes establishing a time-frequency shifting matrix of dimension N×N, wherein N is greater than one. The method further includes combining the time-frequency shifting matrix with a data frame to provide an intermediate data frame. A transformed data matrix is provided by permuting elements of the intermediate data frame. A modulated signal is generated in accordance with elements of the transformed data matrix.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.14/717,886, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMALTIMEFREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed on May 20, 2015,which is a continuation of U.S. patent application Ser. No. 13/927,088,entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIMEFREQUENCYSHIFTING COMMUNICATIONS SYSTEM, filed on Jun. 25, 2013, which claims thebenefit of priority under 35 U.S.C. § 119(e) of U.S. ProvisionalApplication Ser. No. 61/664,020, entitled MODULATION AND EQUALIZATION INAN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Jun.25, 2012, of U.S. Provisional Application Ser. No. 61/801,398, entitledMODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTINGCOMMUNICATIONS SYSTEM, filed Mar. 15, 2013, of U.S. ProvisionalApplication Ser. No. 61/801,366, entitled MODULATION AND EQUALIZATION INAN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar.15, 2013, of U.S. Provisional Application Ser. No. 61/801,435, entitledMODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTINGCOMMUNICATIONS SYSTEM, filed Mar. 15, 2013, of U.S. ProvisionalApplication Ser. No. 61/801,495, entitled MODULATION AND EQUALIZATION INAN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar.15, 2013, of U.S. Provisional Application Ser. No. 61/801,994, entitledMODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTINGCOMMUNICATIONS SYSTEM, filed Mar. 15, 2013, and of U.S. ProvisionalApplication Ser. No. 61/801,968, entitled MODULATION AND EQUALIZATION INAN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar.15, 2013, the contents of each of which are hereby incorporated byreference in their entirety for all purposes. This application is acontinuation-in-part of U.S. patent application Ser. No. 13/117,119,entitled ORTHONORMAL TIME-FREQUENCY SHIFTING AND SPECTRAL SHAPINGCOMMUNICATIONS METHOD, filed May 26, 2011, which claims priority to U.S.Provisional Application Ser. No. 61/349,619, entitled ORTHONORMALTIME-FREQUENCY SHIFTING AND SPECTRAL SHAPING COMMUNICATIONS METHOD,filed May 28, 2010, and is a continuation-in-part of U.S. patentapplication Ser. No. 13/117,124, entitled COMMUNICATIONS METHODEMPLOYING ORTHONORMAL TIME-FREQUENCY SHIFTING AND SPECTRAL SHAPING,filed May 26, 2011, which claims priority to U.S. ProvisionalApplication Ser. No. 61/349,619, entitled ORTHONORMAL TIME-FREQUENCYSHIFTING AND SPECTRAL SHAPING COMMUNICATIONS METHOD, filed May 28, 2010,the contents of each of which are hereby incorporated by reference intheir entirety for all purposes.

FIELD

This disclosure generally relates communications protocols and methods,and more particularly relates to methods for modulation and relatedprocessing of signals used for wireless and other forms ofcommunication.

BACKGROUND

Modern electronic communication devices, such as devices configured tocommunicate over transmission media such as optical fibers, electronicwires/cables, or wireless links, all operate by modulating signals andsending these signals over the applicable transmission medium. Thesesignals, which generally travel at or near the speed of light, can besubjected to various types of degradation or channel impairments. Forexample, echo signals can potentially be generated by optical fiber orwire/cable mediums whenever the modulated signal encounters junctions inthe optical fiber or wire/cable. Echo signals can also potentially begenerated when wireless signals bounce off of wireless reflectingsurfaces, such as the sides of buildings, and other structures.Similarly, frequency shifts can occur when the optical fiber orwire/cable pass through different regions of fiber or cable withsomewhat different signal propagating properties or different ambienttemperatures. For wireless signals, signals transmitted to or from amoving vehicle can encounter Doppler effects that also result infrequency shifts. Additionally, the underlying equipment (i.e.transmitters and receivers) themselves do not always operate perfectly,and can produce frequency shifts as well.

These echo effects and frequency shifts are undesirable and, if suchshifts become too large, may adversely affect network performance byeffectively lowering maximum attainable data rates and/or increasingerror rates. Such performance degradation is particularly problematic inwireless networks, which are straining to accommodate more and moreusers, each desiring to send and receive ever-increasing amounts ofdata. Within wireless networks, the adverse effects arising from echoeffects and frequency shifts stem at least in part from thecharacteristics of existing wireless devices having wirelesscommunication capability. In particular, these portable wireless devices(such as cell phones, portable computers, and the like) are oftenpowered by small batteries, and the users of such devices typicallyexpect to operate them for many hours before recharging is required. Tomeet these user expectations, the wireless transmitters on these devicesmust output wireless signals using very small amounts of power, makingit difficult to distinguish the wireless radio signal over backgroundnoise.

An additional problem is that many of these devices are carried onmoving vehicles, such as automobiles. This causes additionalcomplications because the low-power wireless signal transmitted by thesedevices can also be subjected to various distortions, such as varyingand unpredictable Doppler shifts, and unpredictable multi-path effectsoften caused by varying radio reflections off of buildings or otherstructures.

Moreover, the noise background of the various wireless channels becomesever higher as noise-producing electrical devices proliferate. Theproliferation of other wireless devices also adds to the backgroundnoise.

SUMMARY

The system and method for wideband communications disclosed herein iscapable of operating using relatively low amounts of power whilemaintaining improved resistance to problems of Doppler shift, multi-pathreflections, and background noise. Although examples in the context ofwireless communications will be used throughout this application, themethods disclosed herein are intended, unless stated otherwise, to beequally applicable to wired communication systems.

In one aspect, the present disclosure describes a method of providing amodulated signal useable in a signal transmission system. The method ofthis aspect includes establishing an original data frame having a firstdimension of at least N elements and a second dimension of at least Nelements, wherein N is greater than one. The method further includestransforming the original data frame in accordance with a time-frequencytransformation so as to provide a transformed data matrix. A modulatedsignal is generated in accordance with elements of the transformed datamatrix.

In another aspect, the present disclosure describes a method formodulating data for transmission within a communication system. Themethod of this aspect includes establishing a time-frequency shiftingmatrix of dimension N×N, wherein N is greater than one. The methodfurther includes combining the time-frequency shifting matrix with adata frame to provide an intermediate data frame. A transformed datamatrix is provided by permuting elements of the intermediate data frame.A modulated signal is generated in accordance with elements of thetransformed data matrix.

In another aspect, the present disclosure relates to a signaltransmitter for use in a communication system. The signal transmitterincludes an input port, an output port, a processor, and a memoryincluding program code executable by the processor. The program codeincludes code for receiving input data at the input port. The programcode further includes code for establishing, using the input data, anoriginal data frame having a first dimension of at least N elements anda second dimension of at least N elements, wherein N is greater thanone. In addition, the program code includes code for transforming theoriginal data frame in accordance with a time-frequency transformationso as to provide a transformed data matrix and code for generating amodulated signal in accordance with elements of the transformed datamatrix.

In a further aspect the present disclosure pertains to a signaltransmitter for use in a communication system. The signal transmitterincludes an input port, an output port, a processor, and a memoryincluding program code executable by the processor. The program codeincludes code for receiving, at the input port, input data. The programcode further includes code for establishing a time-frequency shiftingmatrix of dimension N×N, wherein N is greater than one. In addition, theprogram code includes code for combining the time-frequency shiftingmatrix with a data frame to provide an intermediate data frame whereinthe data frame includes the input data. Code for providing a transformeddata matrix by permuting elements of the intermediate data frame andcode for generating a modulated signal in accordance with elements ofthe transformed data matrix is also provided.

The present disclosure is further directed to a non-transitory computerreadable medium including program instructions for execution by aprocessor in a signal transmitter. The program instructions includeinstructions for causing the processor to receive, at an input port ofthe signal transmitter, input data and establish, using the input data,an original data frame having a first dimension of at least N elementsand a second dimension of at least N elements, wherein N is greater thanone. The instructions further cause the processor to transform theoriginal data frame in accordance with a time-frequency transformationso as to provide a transformed data matrix and generate a modulatedsignal in accordance with elements of the transformed data matrix.

In yet another aspect the present disclosure pertains to anon-transitory computer readable medium including program instructionsfor execution by a processor in a signal transmitter. The programinstructions include instructions for causing the processor to receive,at an input port of the signal transmitter, input data and establish atime-frequency shifting matrix of dimension N×N, wherein N is greaterthan one. The instructions further cause the processor to combine thetime-frequency shifting matrix with a data frame to provide anintermediate data frame wherein the data frame includes the input dataand to provide a transformed data matrix by permuting elements of theintermediate data frame. In addition, the instructions includeinstructions which cause the processor to generate a modulated signal inaccordance with elements of the transformed data matrix.

In another aspect, the present disclosure describes a method ofreceiving data. The method of this aspect includes receiving datasignals corresponding to a transmitted data frame comprised of a set ofdata elements, constructing, based upon the data signals, a receiveddata frame having a first dimension of at least N elements and a seconddimension of at least N elements, where N is greater than one, inversepermuting at least a portion of the elements of the received data frameso as to form an non-permuted data frame, and inverse transforming thenon-permuted data frame in accordance with first inverse-transformationmatrix so as to form a recovered data frame corresponding to areconstructed version of the transmitted data frame.

In another aspect, the present disclosure describes a method of datatransmission. The method of this aspect includes arranging a set of dataelements into an original data frame having a first dimension of Nelements and a second dimension of N elements, where N is greater thanone, and transforming the original data frame in accordance with atransformation matrix to form a first transformed data matrix having atleast N² transformed data elements wherein each of the transformed dataelements is based upon a plurality of the data elements of the originaldata frame and wherein a first dimension of the first transformed datamatrix corresponds to a frequency shift axis and a second dimensioncorresponds to a time shift axis. The method of this aspect furtherincludes forming a permuted data matrix by permuting at least a portionof the elements of the first transformed data matrix so as to shift theat least a portion of the elements with respect to the time shift axis,transforming the permuted data matrix using a frequency-shift encodingmatrix to form a transmit frame, and generating a modulated signal inaccordance with elements of the transmit frame.

In another aspect, the present disclosure describes a method ofreceiving data. The method of this aspect includes receiving, on one ormore carrier waveforms, signals representing a plurality of dataelements of an original data frame wherein each of the data elements arerepresented by cyclically time shifted and cyclically frequency shiftedversions of a known set of waveforms, generating, based upon thesignals, a received data frame having a first dimension of at least Nelements and a second dimension of at least N elements, where N isgreater than one, wherein the first dimension corresponds to a frequencyshift axis and the second dimension corresponds to a time shift axis.The method of this aspect further includes performing, using a decodingmatrix, an inverse transformation operation with respect to elements ofthe received data frame so as to yield a non-transformed matrix, andgenerating, based upon the non-transformed matrix, a recovered dataframe comprising an estimate of the original data frame.

In another aspect, the present disclosure describes a new signalmodulation technique which involves spreading data symbols over a largerange of times, frequencies, and/or spectral shapes (waveforms). Suchmethod, termed “Orthonormal Time-Frequency Shifting and Spectral Shaping(“OTFSSS”) when spectral shaping is employed or, more generally, “OTFS”,operates by sending data in what are generally substantially larger“chunks” or frames than the data frames used in prior methods. That is,while prior art methods might encode and send units or frames of “N”symbols over a communications link over a particular time interval, OTFScontemplates that frames of N² symbols are transmitted (often over arelatively longer time interval). With OTFS modulation, each data symbolor element that is transmitted is extensively spread in a novel mannerin time, frequency and/or spectral shape space. At the receiving end ofthe connection, each data symbol is resolved based upon substantiallythe entire frame of N² received symbols.

In another aspect, disclosed herein is a wireless communication methodpredicated upon spreading input data over time, frequency andpotentially spectral shape using convolution unit matrices (data frames)of N×N (N²). In general, either all N² of the data symbols are receivedover N particular spreading time intervals (each composed of N timeslices), or no such symbols are received. During the transmissionprocess, each N×N data frame matrix will typically be multiplied by afirst N×N time-frequency shifting matrix, permuted, and then multipliedby a second N×N spectral-shaping matrix, thereby mixing each data symbolacross the entire resulting N×N matrix (which may be referred to as theTFSSS data matrix). Columns from the TFSSS data matrix are thenselected, modulated, and transmitted, on a one element per time slicebasis. At the receiver, a replica TFSSS matrix is reconstructed anddeconvolved, yielding a reconstruction of the input data.

Embodiments of the systems and methods described herein may use noveltime-frequency shifting and spectral shaping codes to spread data acrosstime, spectrum, waveform, and/or spectral-shape. In such embodimentstime-shifting techniques, frequency-shifting techniques and, optionally,spectral-shaping techniques may be used in conjunction to transmit dataat high rates in a manner unusually resistant to problems caused byDoppler shifts, multi-path effects, and background noise.

During the signal transmission process, an OTFS transmitter maysubdivide and transmit each data element or symbol over a cyclicallyvarying range of frequencies and over a series of spreading timeintervals. Often this will require that each data element or symbol betransmitted over a somewhat longer period of time than is utilized fortransmission data frames in other communication systems. Notwithstandingthese potentially longer transmission periods, the OTFS system iscapable of achieving superior data rate performance by using a complexmultiplexing methods premised on the convolution and deconvolutionschemes discussed herein. Through use of such methods a comparativelylarge amount of information may be included within each transmittedsignal. In particular, the relatively large number (i.e., N²) of datasymbols or elements transmitted during each data frame using theconvolution and deconvolution schemes disclosed herein enablecomparatively high data rates to be attained despite the diminution indata rate which could otherwise result from division of a single dataelement or symbol over N time-spreading intervals. Moreover, becauseeach data symbol is typically subdivided and sent over a plurality ofsignals, signal processing schemes described herein may be employed topermit data symbols to be recovered even in the event of loss of one ormore of the plurality of transmitted signals. In addition, such schemesmay be employed to compensate for losses due to common wirelesscommunications link impairments, such as Doppler shift and multi-patheffects.

For example, whereas with prior art, if by chance Doppler effects causedby one wireless signal from a first transmitter fall on the samefrequency as another signal from a first or second transmitter (formulti-path effects, a signal from a moving first or second transmitterthat hits an object at an arbitrary angle to the receiver can produce aDoppler distorted reflection or echo signal of the first or secondtransmitter that also reaches the receiver), this could result inconfusion, ambiguity, and data loss. By contrast, by cyclically shiftingthe frequency and sending an element of data over a plurality of timeintervals, the impact of a Doppler “collision” is substantiallyminimized—at most there will be a brief transient effect resulting inthe loss of only one of a plurality of signals used to transmit aparticular data symbol or element. The effects of other communicationslink impairments, such as multi-path effects, can also be minimizedbecause the cyclically shifting frequency provides yet another way tocompensate for multi-path effects.

There exist at least two ways in which a data element or symbol may bepartitioned across a time range of cyclically shifting frequencies, andthus two basic forms of the OTFS method. In a first form of the OTFSmethod, the data from a single symbol is convolved and partitionedacross multiple time slices, and ultimately transmitted as a series oftime slices, on a per time slice basis. When this transmission scheme isused the cyclically-shifting frequency is accomplished over a pluralityof time spreading intervals. Thus, for this first form of the OTFSmethod, the basic unit of data transmission operates on a time slicebasis.

In a second form of the OTFS method, the data is ultimately transmittedas a series of waveforms with characteristic frequencies, where eachwaveform lasts for a spreading interval of time generally consisting ofN time slices. When this transmission scheme is used thecyclically-shifting frequency is accomplished over a plurality of timespreading intervals. Thus, for this second form of the OTFS method, thebasic unit of data transmission operates over a relatively longerspreading time interval comprised of N time slices. Unless otherwisespecified, the discussion within the remainder of this disclosure willfocus on the first form of the OTFS method in which the basic unit ofdata transmission operates on a time slice basis.

Again considering the first form of the OTFS method, in one embodimentthis form of the OTFS method contemplates formation of an N×N data framematrix having N² symbols or elements and multiplication of this dataframe by a first N×N time-frequency shifting matrix. The result of thismultiplication is optionally permuted and, following permutation,optionally multiplied by a second N×N spectral shaping matrix. As aresult, the N² data elements in the frame of data are essentially mixedor distributed throughout the resulting N×N matrix product, here calleda “Time Frequency Shifted” data matrix or “TFS” data matrix. If theoptional spectral shaping is used, the resulting N×N matrix product maybe referred to as a “Time Frequency Shifted and Spectral Shaped” datamatrix or “TFSSS” data matrix. Thus, for example, a single symbol orelement in row 1 column 1 of the N×N frame of data may end up beingdistributed over all rows and columns of the resulting N×N TFS or TFSSSdata matrix (in what follows the term TFS may connote either the TFS orTFSSS data matrix).

The contents (i.e. the individual elements) of this TFS data matrix maythen be selected, modulated, and transmitted. Usually N elements at atime from this TFS data matrix (often a column from the TFS data matrix)are selected to be sent over one spreading interval of time, thus oftenrequiring N spreading intervals of time to transmit the entire contentsof the TFS data matrix. This spreading interval of time in turn isusually composed of at least N time slices. During each time slice, oneelement from the most recent selection of N elements (for example, fromthe selected column of the TFS data matrix) is selected, modulated, andtransmitted.

At the receiving end, the process operates generally in reverse. Theindividual elements of the TFS data matrix are received over varioustime slices and various time spreading intervals, allowing the receiverto reassemble a replica (which may not be a perfect replica due tocommunications link impairment effects) of the original TFS data matrix.Using its knowledge of the first N×N time-frequency shifting matrix, theoptional permutation process, the second N×N spectral shaping matrix,and the selection process used to select different elements of the TFSdata matrix, as well as various noise reduction or compensationtechniques to overcome impairment effects, the receiver will thenreconstruct the original N×N data frame matrix of N² symbols orelements. Because each data symbol or element from the original dataframe is often spread throughout the TFS data matrix, often most or theentire TFS matrix will need to be reconstructed in order to solve forthe original data symbol or element. However, by using noise reductionand compensation techniques, minor data losses during transmission canoften be compensated for.

In some embodiments, advanced signal modulation schemes utilizingcyclically time shifted and cyclically frequency shifted waveforms maybe utilized to correct channel impairments in a broad range ofsituations. For example, in one aspect the OTFS method may contemplatetransfer of a plurality of data symbols using a signal modulated in amanner which effectively compensates for the adverse effects of echoreflections and frequency offsets. This method will generally comprisedistributing this plurality of data symbols into one or more N×N symbolmatrices and using these one or more N×N symbol matrices to control thesignal modulation occurring within a transmitter. Specifically, duringthe transmission process each data symbol within an N×N symbol matrix isused to weight N waveforms. These N waveforms are selected from an N²sized set of all permutations of N cyclically time shifted and Ncyclically frequency shifted waveforms determined according to anencoding matrix U. The net result produces, for each data symbol, Nsymbol-weighted cyclically time shifted and cyclically frequency shiftedwaveforms. Generally this encoding matrix U is chosen to be an N×Nunitary matrix that has a corresponding inverse decoding matrix U^(H).Imposition of this constraint means that the encoding matrix U producesresults which can generally be decoded.

Continuing with this example, for each data symbol in the N×N symbolmatrix, the transmitter may sum the corresponding N symbol-weightedcyclically time shifted and cyclically frequency shifted waveforms, andby the time that the entire N×N symbol matrix is so encoded, produce N²summation-symbol-weighted cyclically time shifted and cyclicallyfrequency shifted waveforms. The transmitter will then transmit these N²summation-symbol-weighted cyclically time shifted and cyclicallyfrequency shifted waveforms, structured as N composite waveforms, overany combination of N time blocks or frequency blocks.

To receive and decode this transmission, the transmitted N²summation-symbol-weighted cyclically time shifted and cyclicallyfrequency shifted waveforms are subsequently received by a receiverwhich is controlled by the corresponding decoding matrix U^(H). Thereceiver will then use this decoding matrix U^(H) to reconstruct theoriginal symbols in the various N×N symbol matrices.

This process of transmission and reception will normally be performed byvarious electronic devices, such as a microprocessor equipped, digitalsignal processor equipped, or other electronic circuit that controls theconvolution and modulation parts of the signal transmitter. Similarlythe process of receiving and demodulation will also generally rely upona microprocessor equipped, digital signal processor equipped, or otherelectronic circuit that controls the demodulation, accumulation, anddeconvolution parts of the signal receiver. However, although theexemplary techniques and systems disclosed herein will often bediscussed within the context of a wireless communication systemcomprised of at least one wireless transmitter and receiver, it shouldbe understood that these examples are not intended to be limiting. Inalternative embodiments, the transmitter and receiver may beoptical/optical-fiber transmitters and receivers, electronic wire orcable transmitters and receivers, or other types of transmitters inreceivers. In principle, more exotic signal transmission media, such asacoustic signals and the like, may also be utilized in connection withthe present methods.

As previously discussed, regardless of the media (e.g. optical,electrical signals, or wireless signals) used to transmit the variouswaveforms, these waveforms can be distorted or impaired by varioussignal impairments such as various echo reflections and frequencyshifts. As a result, the receiver will often receive a distorted form ofthe original signal. Here, embodiments of the OTFS method make use ofthe insight that cyclically time shifted and cyclically frequencyshifted waveforms are particularly useful for detecting and correctingfor such distortions.

Because communications signals propagate through their respectivecommunications media at a finite speed (often at or near the speed oflight), and because the distance from a transmitter to a receiver isusually substantially different than the distance between thetransmitter, the place(s) where the echo is generated, and the distancebetween the place(s) where the echo is generated and the receiver, thenet effect of echo reflections is that both an originally transmittedwaveform and time-shifted versions thereof are received at the receiver,thereby resulting in a distorted composite signal. However, embodimentsof the OTFS method utilizing cyclically time shifted waveforms may beemployed to counteract this distortion. In particular, a timedeconvolution device at the receiver may operate to analyze thecyclically time varying patterns of these waveforms, determine therepeating patterns, and use these repeating patterns to help decomposethe echo distorted signal back into various time-shifted versions of thevarious signals. The time deconvolution device can also determine howmuch of a time-offset (or multiple time offsets) is or are required toenable the time delayed echo signal(s) to match up with the originallytransmitted signal. This time offset value, which may be referred toherein as a time deconvolution parameter, can both give usefulinformation as to the relative position of the echo location(s) relativeto the transmitter and receiver, and can also help the systemcharacterize some of the signal impairments that occur between thetransmitter and receiver. This can help the communications systemautomatically optimize itself for better performance.

In addition to echo reflections, other signal distortions may occur thatcan result in one or more frequency shifts. For example, a Doppler shiftor Doppler effects can occur when a wireless mobile transmitter movestowards or away from a stationary receiver. If the wireless mobiletransmitter is moving towards the stationary receiver, the wirelesswaveforms that it transmits will be offset to higher frequencies, whichcan cause confusion if the receiver is expecting signals modulated at alower frequency. An even more confusing result can occur if the wirelessmobile transmitter is moving perpendicular to the receiver, and there isalso an echo source (such as a building) in the path of the wirelessmobile transmitter. Due to Doppler effects, the echo source receives ablue shifted (higher frequency) version of the original signal, andreflects this blue shifted (higher frequency) version of the originalsignal to the receiver. As a result, the receiver will receive both theoriginally transmitted “direct” wireless waveforms at the original lowerfrequency, and also a time-delayed higher frequency version of theoriginal wireless waveforms, causing considerable confusion.

It has been found that the use of cyclically time shifted waveforms andcyclically frequency shifted waveforms may help address this type ofproblem. In particular, it has been found that the cyclic variationyields important pattern matching information which may enable thereceiver to determine which portions of a received signal have beendistorted and the extent of such distortion. In one embodiment thesecyclically varying signals allow the receiver to perform atwo-dimensional (e.g. time and frequency) deconvolution of the receivedsignal. For example, the frequency deconvolution portion of the receivercan analyze the cyclically frequency varying patterns of the waveforms,essentially do frequency pattern matching, and decompose the distortedsignal into various frequency shifted versions of the various signals.At the same time, this portion of the receiver can also determine howmuch of a frequency offset is required to cause the frequency distortedsignal to match up with the originally transmitted signal. Thisfrequency offset value, herein termed a “frequency deconvolutionparameter”, can give useful information as to the transmitter's velocityrelative to the receiver. This may facilitate characterization of someof the frequency shift signal impairments that occur between thetransmitter and receiver.

As before, the time deconvolution portion of the receiver can analyzethe cyclically time varying patterns of the waveforms, again do timepattern matching, and decompose the echo distorted signal back intovarious time-shifted versions of the original signal. The timedeconvolution portion of the receiver can also determine how much of atime-offset is required to cause the time delayed echo signal to matchup with the original or direct signal. This time offset value, againcalled a “time deconvolution parameter”, can also give usefulinformation as to the relative positions of the echo location(s), andcan also help the system characterize some of the signal impairmentsthat occur between the transmitter and receiver.

The net effect of both the time and frequency deconvolution, whenapplied to transmitters, receivers, and echo sources that potentiallyexist at different distances and velocities relative to each other, isto allow the receiver to properly interpret the impaired echo andfrequency shifted communications signals.

Further, even if, at the receiver, the energy received from theundistorted form of the originally transmitted signal is so low as tohave a undesirable signal to noise ratio, by applying the appropriatetime and frequency offsets or deconvolution parameters, the energy fromthe time and/or frequency shifted versions of the signals (which wouldotherwise be contributing to noise) can instead be harnessed tocontribute to the signal instead.

As before, the time and frequency deconvolution parameters can alsoprovide useful information as to the relative positions and velocitiesof the echo location(s) relative to the transmitter and receiver, aswell as the various velocities between the transmitter and receiver.These in turn can help the system characterize some of the signalimpairments that occur between the transmitter and receiver, as well asassist in automatic system optimization methods.

Thus in some embodiments, the OTFS system may also provide a method foran improved communication signal receiver where, due to either one orthe combination of echo reflections and frequency offsets, multiplesignals due to echo reflections and frequency offsets result in thereceiver receiving a time and/or frequency convolved signal representingtime and/or frequency shifted versions of the N²summation-symbol-weighed cyclically time shifted and frequency shiftedwaveforms previously sent by the transmitter. Here, the improvedreceiver will further perform a time and/or frequency deconvolution ofthe impaired signal to correct for various echo reflections andfrequency offsets. This improved receiver method will result in bothtime and frequency deconvolved results (i.e. signals with higher qualityand lower signal to noise ratios), as well as various time and frequencydeconvolution parameters that, in addition to automatic communicationschannel optimization, are also useful for other purposes as well. Theseother purposes can include channel sounding (i.e. better characterizingthe various communication system signal impairments), adaptivelyselecting modulation methods according to the various signalimpairments, and even improvements in radar systems.

BRIEF DESCRIPTION OF THE DRAWINGS

For a better understanding of the nature and objects of variousembodiments of the invention, reference should be made to the followingdetailed description taken in conjunction with the accompanyingdrawings, wherein:

FIG. 1 illustrates an example of a wireless communication system thatmay exhibit time/frequency selective fading.

FIG. 2 shows an exemplary mathematical model that can be used to modelcommunication in the wireless communication system of FIG. 1.

FIG. 3A shows an exemplary block diagram of components of an OTFScommunication system.

FIG. 3B illustrates a process by which an OTFS transceiver of atransmitting device within the system of FIG. 3A may transmit a dataframe.

FIG. 3C illustrates a process by which an OTFS transceiver of areceiving device within the system of FIG. 3A may operate to receive atransmitted data frame.

FIG. 4A illustrates components of an exemplary OTFS transceiver.

FIG. 4B illustrates an exemplary process pursuant to which an OTFStransceiver may transmit, receive and reconstruct information utilizinga TFS data matrix.

FIG. 5 illustrates a comparison of predicted bit error rate between anexemplary OTFS method and a time division multiple access method over anexemplary communication channel exhibiting time/frequency fading.

FIG. 6A shows an overview of one manner in which an OTFS method may beused to transmit data over a wireless link.

FIG. 6B illustrates components of an exemplary OTFS transmitter forperforming the method of FIG. 6A.

FIG. 6C is a flowchart representative of an exemplary OTFS datatransmission method.

FIG. 7A shows an overview of one manner in which an OTFS method may beused to receive data over a wireless link.

FIG. 7B illustrates components of an exemplary OTFS receiver forperforming the method of FIG. 7A.

FIG. 7C is a flowchart representative of an exemplary OTFS datademodulation method.

FIG. 8 shows an exemplary set of basic building blocks used to convolveand deconvolve data according to a second form of the OTFS method.

FIG. 9 shows an exemplary transmit frame including guard times betweengroups of transmitted data.

FIG. 10 shows a diagram of the cyclic convolution method used toconvolve data and transmit data according to the second form of the OTFSmethod.

FIG. 11 shows an exemplary structure of a receive frame resulting fromthe transmit frame of FIG. 9.

FIG. 12 shows a diagram of the cyclic deconvolution method used todeconvolve received data according to the second form of the OTFSmethod.

FIGS. 13A and 13B illustrate operations performed by a transmitterconsistent with a first alternative OTFS transmission scheme.

FIGS. 14A and 14B illustrate operations performed by a receiverconsistent with the first alternative OTFS scheme.

FIGS. 15A and 15B illustrate operations performed by a transmitterconsistent with a second alternative OTFS scheme.

FIGS. 16A and 16B illustrate operations performed by a receiverconsistent with the second alternative OTFS scheme.

FIG. 17 illustrates a unitary matrix [U1] in the form of an identitymatrix representative of a time division multiplexed transmission basis.

FIG. 18 illustrates a unitary matrix [U1] in the form of a DFT matrixrepresentative of a frequency division multiplexed transmission basis.

FIG. 19 illustrates a unitary matrix [U1] in the form of a Hadamardmatrix representative of a code division multiplexed transmission basis.

FIG. 20 illustrates a sequence of L-OTFS N×N matrices making up a frameof data comprising L×N×N symbols spread in both time and frequency.

FIG. 21A shows a more detailed diagram of one embodiment of an OTFStransmitter module.

FIG. 21B depicts a TFS matrix generated within the OTFS transmitter ofFIG. 21A.

FIG. 21C depicts a timeline relevant to operation of the transmitter ofFIG. 21A.

FIG. 22 illustrates an exemplary permutation operation that can be usedin an OTFS modulation scheme.

FIG. 23 illustrates another exemplary permutation operation that can beused in an OTFS modulation scheme.

FIG. 24 illustrates a first exemplary time and frequency tiling approachthat can be used in an OTFS modulation scheme.

FIG. 25 illustrates a second exemplary time and frequency tilingapproach that can be used in an OTFS scheme.

FIG. 26 illustrates a third exemplary time and frequency tiling schemethat can be used in the OTFS modulation scheme.

FIG. 27 illustrates the transmission of cyclically time shiftedwaveforms in order to enable time deconvolution of the received signalto be performed so as to compensate for various types of echoreflections.

FIG. 28 illustrates the transmission of both cyclically time shiftedwaveforms and cyclically frequency shifted waveforms in order to enableboth time and frequency deconvolution of the received signal to beperformed so as to compensate for both echo reflections and frequencyshifts.

FIG. 29 illustrates the transmission of various composite waveformblocks as either a series of N consecutive time blocks associated withina single symbol matrix or, alternatively, as a time-interleaved seriesof blocks from different symbol matrices.

FIG. 30 illustrates the transmission of various composite waveformblocks as either shorter duration time blocks over one or more widerfrequency ranges or as longer duration time blocks over one or morenarrower frequency ranges.

FIG. 31 shows a high-level representation of a receiver processingsection configured to mathematically compensate for the effects of echoreflections and frequency shifts using an equalizer.

FIG. 32A shows an example of a communications channel in which echoreflections and frequency shifts can blur or impair or distort atransmitted signal.

FIG. 32B shows an example of an adaptive linear equalizer that may beused to correct for distortions.

FIG. 32C shows an example of an adaptive decision feedback equalizerthat may be used to correct for distortions.

FIG. 33 shows a time-frequency graph illustrating the various echo (timeshifts) and frequency shifts that a signal may encounter duringpropagation through a channel.

FIG. 34 illustratively represents a time-frequency map of tap valuesproduced by the feed forward (FF) portion of the adaptive decisionfeedback equalizer.

FIG. 35 illustratively represents a time-frequency map of tap valuesproduced by the feedback (FB) portion of the adaptive decision feedbackequalizer.

FIGS. 36A and 36B demonstrate the utility of transmitting variousdifferent time blocks consistent with an interleaving scheme based atleast in part upon expected latency.

FIG. 37 illustrates an example of a full duplex OTFS transceiver inaccordance with the disclosure.

FIG. 38 illustrates an example of an OTFS receiver that providesiterative signal separation in accordance with the disclosure.

FIG. 39A is a basis matrix including N length N basis vectors b₀-b_(N−1)used in illustrating one manner in which OTFS encoding using a pair oftransform matrices or frames can spread N² data symbols d_(ij) into N²different basis matrices B_(ij) of basis frames F_(ij).

FIG. 39B illustrates an incomplete basis matrix that includes N-lcolumns and N-k rows where l and k are greater than or equal to one.

FIG. 39C illustrates a basis frame that has N vectors of length M whereM is greater than N.

FIG. 39D illustrates an incomplete basis frame including N-l columns andM-k rows, where l and k are greater than or equal to one.

FIG. 40 is a block diagram of a time-frequency-space decision feedbackequalizer that may be employed to facilitate signal separation in amulti-antenna OTFS system.

FIG. 41 is a block diagram of a time-frequency-space decisionfeedforward FIR filter.

FIG. 42 is a block diagram of a time-frequency-space decision feedbackFIR filter.

FIG. 43 provides a high-level representation of a conventionaltransceiver which could be utilized in an exemplary wirelesscommunication system.

FIGS. 44A and 44B provide block diagrammatic representations ofembodiments of first and second OTFS transceivers configured to utilizea spreading kernel.

FIG. 45 is a flowchart representative of the operations performed by anOTFS transceiver.

FIG. 46 illustrates the functioning of a modulator as an orthogonal mapdisposed to transform a two-dimensional time-frequency matrix to atransmitted waveform.

FIGS. 47 and 48 illustrate the transformation by a demodulator of areceived waveform to a two-dimensional time-frequency matrix inaccordance with an orthogonal map.

FIG. 49 illustrates an exemplary implementation of a two-dimensionaldecision feedback equalizer configured to perform a least means square(LMS) equalization procedure.

FIG. 50 shows an OTFS mesh network within the context of a cellularcommunication system comprised of cell sites and associated cellcoverage areas.

FIG. 51 shows an OTFS mesh network organized around a set of wirednetwork gateways.

FIG. 52 shows an OTFS mesh network system comprised of a single-channelwireless mesh network including a plurality of mesh elements.

FIG. 53 provides an illustration of a two-dimensional channel impulse.

FIGS. 54A-54C depict input and output streams after two-dimensionalchannel distortion.

DETAILED DESCRIPTION

One unique aspect of the signal modulation techniques described hereinis the concept of spreading the data of a single symbol over arelatively large range of times, frequencies, and spectral shapes. Incontrast, prior communication systems have been predicated uponassigning a given data symbol to a specific time-spreading interval ortime slice uniquely associated with such data symbol. As is discussedbelow, the disclosed OTFS method is based at least in part upon therealization that in many cases various advantages may accrue fromspreading the data of a single symbol over multiple time-spreadingintervals shared with other symbols. In contrast with prior artmodulation techniques, the OTFS method may involve convolving a singledata symbol over both a plurality of time slots, a plurality offrequencies or spectral regions (spread spectrum), and a plurality ofspectral shapes. As is described below, this approach to dataconvolution results in superior performance over impaired communicationslinks.

System Overview

FIG. 1 illustrates an example of a wireless communication system 100that may exhibit time/frequency selective fading. The system 100includes a transmitter 110 (e.g., a cell phone tower) and a receiver 120(e.g., a cell phone). The scenario illustrated in FIG. 1 includesmultiple pathways (multi-path) that the signal transmitted from thetransmitter 100 travels through before arriving at the receiver 100. Afirst pathway 130 reflects through a tree 132, second pathway 140reflects off of a building 142 and a third pathway 150 reflects off of asecond building 152. A fourth pathway 160 reflects off of a moving car162. Because each of the pathways 130, 140, 150 and 160 travels adifferent distance, and is attenuated or faded at a different level andat a different frequency, when conventionally configured the receiver120 may drop a call or at least suffer low throughput due to destructiveinterference of the multi-path signals.

Turning now to FIG. 43, a high-level representation is provided of aconventional transceiver 4300 which could be utilized in the wirelesscommunication system 100 of FIG. 1. The transceiver 4300 could, forexample, operate in accordance with established protocols fortime-division multiple access (TDMA), code-division multiple access(CDMA) or orthogonal frequency-division multiple access (OFDM) systems.In conventional wireless communication systems such as TDMA, CDMA, andOFDM) systems, the multipath communication channel 4310 between atransmitter 4304 and a receiver 4308 is represented by a one-dimensionalmodel. In these systems channel distortion is characterized using aone-dimensional representation of the impulse response of thecommunication channel. The transceiver 4300 may include aone-dimensional equalizer 4320 configured to at least partially removethis estimated channel distortion from the one-dimensional output datastream 4330 produced by the receiver 4308.

Unfortunately, use of a one-dimensional channel model presents a numberof fundamental problems. First, the one-dimensional channel modelsemployed in existing communication systems are non-stationary; that is,the symbol-distorting influence of the communication channel changesfrom symbol to symbol. In addition, when a channel is modeled in onlyone dimension it is likely and possible that certain received symbolswill be significantly lower in energy than others due to “channelfading”. Finally, one-dimensional channel state information (CSI)appears random and much of it is estimated by interpolating betweenchannel measurements taken at specific points, thus rendering theinformation inherently inaccurate. These problems are only exacerbatedin multi-antenna (MIMO) communication systems. As is discussed below,embodiments of the OTFS method described herein can be used tosubstantially overcome the fundamental problems arising from use of aone-dimensional channel model.

As is indicated below by Equation (1), in one aspect the OTFS methodrecognizes that a wireless channel may be represented as a weightedsuperposition of combination of time and Doppler shifts:

In contrast to the parameters associated with existing channel models,the time-frequency weights (τ,) of Equation (1) are two-dimensional andare believed to fully characterize the wireless channel. Thetime-frequency weights (τ,) are intended to represent essentially all ofthe diversity branches existing in the wireless channel. This isbelieved to substantially minimize the fading effects experienced by theOTFS system and other communication systems generally based upontwo-dimensional channel models relative to the fading common in systemspredicated upon one-dimensional models. Finally, in contrast to thenon-stationary, one-dimensional channel models employed in conventionalcommunication systems, the time-frequency weights (τ,) of Equation (1)are substantially stationary; that is, the weights change very slowlyrelative to the time scale of exemplary embodiments of the OTFS system.

Use of the two-dimensional channel model of Equation (1) in embodimentsof the OTFS communication system affords a number of advantages. Forexample, use of the channel model of Equation (1) enables both channelmultipath delay and Doppler shift to be accurately profiledsimultaneously. Use of this model and the OTFS modulation techniquesdescribed herein also facilitate the coherent assembly of channel echoesand the minimization of fading phenomena, since every symbol experiencesubstantially all of the diversity branches present within the channel.Given that the two-dimensional channel model is essentially stationary,every symbol is deterministically distorted (smeared) according tosubstantially the same two-dimensional pattern. This stable, accuratecharacterization of the communication channel in two dimensions on anongoing basis further enables the OTFS system to minimize datadistortion by “customizing” how each bit is delivered across thechannel. Finally, use of a two-dimensional channel model enableseffective signal separation by decoupling and eliminating mutualinterference between multiple sources.

Attention is now directed to FIG. 2, which illustrates an example of amathematical model 200 that can be used to model time/frequencyselective fading. A transmit side of the model 200 includes apre-equalizer 210, a transmitter/modulation component 220, a channelmodel 230, and additive noise 240 which is combined with the transmittedsignal via a summer 250. A receive side of the model 200 includes areceiver/demodulator 260 and a post equalizer 270.

The pre-equalizer 210 is used to model a pre-distortion transferfunction h_(t) that can be used to make up for changing channelconditions in the channel model h_(c) based on feedback received overthe channel from the receive side of the model, as determined bymeasurements made by the receiver/demodulator 260 and/or the postequalizer 270. The transmitter/modulator 220 uses modulation schemesdescribed herein to transmit the data over the channel 230.

The receiver/demodulator 260 demodulates the signal received over thechannel 230. The received signal has been distorted by time/frequencyselective fading, as determined by the channel transfer function h_(c),and includes the additive noise 240. The receiver/demodulator 260 andthe post equalizer 270 utilize methods discussed herein to reduce thedistortion caused by the time/frequency selective fading and additivenoise due to the channel conditions. The mathematical model 200 can beused to determine the nature of the equalized data D_(eq) by performinga mathematical combination of three transfer functions operating on theoriginal data D. The three transfer functions include the transmittertransfer function h_(t), the channel transfer function h_(c) and theequalizer transfer function hr.

Embodiments of the OTFS methods and systems described herein are based,in part, upon the realization that spreading the data for any givensymbol over time, spectrum, and/or spectral shapes in the mannerdescribed herein yields modulated signals which are substantiallyresistant to interference, particularly interference caused by Dopplereffects and multi-path effects, as well as general background noiseeffects. Moreover, the OTFS method is believed to require less precisefrequency synchronization between receiver and transmitter than isrequired by existing communication systems (e.g., OFDM systems).

In essence, the OTFS method convolves the data for a group of N² symbols(herein called a “frame”) over both time, frequency, and in someembodiments spectral shape in a way that results in the data for thegroup of symbols being sent over a generally longer period of time thanin prior art methods. Use of the OTFS method also results in the datafor any given group of symbols being accumulated over a generally longerperiod of time than in prior art methods. However, in certainembodiments the OTFS method may nonetheless enable favorable data ratesto be achieved despite the use of such longer transmission periods byexploiting other transmission efficiencies enabled by the method. Forexample, in one embodiment a group of symbols may be transmitted usingthe same spread-spectrum code. Although this could otherwise result inconfusion and ambiguity (since each symbol would not be uniquelyassociated with a code), use of the OTFS method may, for example, enablethe symbols to be sent using different (but previously defined)spread-spectrum convolution methods across a range of time and frequencyperiods. As a consequence, when all of the data corresponding to thesymbols is finally accumulated within the receiver, the entire frame orgroup of symbols may be reconstructed in a manner not contemplated byprior art techniques. In general, one trade-off associated with thedisclosed approach is that either an entire multi-symbol frame of datawill be correctly received, or none of the frame will be correctlyreceived; that is, if there is too much interference within thecommunication channel, then the ability to successfully deconvolve andretrieve multiple symbols may fail. However, as will be discussed,various aspects of the OTFS may mitigate any degradation in performancewhich would otherwise result from this apparent trade-off.

FIG. 3A is a block diagram of components of an exemplary OTFScommunication system 300. As shown, the system 300 includes atransmitting device 310 and a receiving device 330. The transmittingdevice 310 and the receiving device 330 include first and second OTFStransceivers 315-1 and 315-2, respectively. The OTFS transceivers 315-1and 315-2 communicate, either unidirectionally or bidirectionally, viacommunication channel 320 in the manner described herein. Although inthe exemplary embodiments described herein the system 300 may comprise awireless communication system, in other embodiments the communicationchannel may comprise a wired communication channel such as, for example,a communication channel within a fiber optic or coaxial cable. As wasdescribed above, the communication channel 320 may include multiplepathways and be characterized by time/frequency selective fading.

FIG. 4 illustrates components of an exemplary OTFS transceiver 400. TheOTFS transceiver 400 can be used as one or both of the exemplary OTFStransceivers 315 illustrated in the communication system 300 of FIG. 3.The OTFS transceiver 400 includes a transmitter module 405 that includesa pre-equalizer 410, an OTFS encoder 420 and an OTFS modulator 430. TheOTFS transceiver 400 also includes a receiver module 455 that includes apost-equalizer 480, an OTFS decoder 470 and an OTFS demodulator 460. Thecomponents of the OTFS transceiver may be implemented in hardware,software, or a combination thereof. For a hardware implementation, theprocessing units may be implemented within one or more applicationspecific integrated circuits (ASICs), digital signal processors (DSPs),digital signal processing devices (DSPDs), programmable logic devices(PLDs), field programmable gate arrays (FPGAs), processors, controllers,microcontrollers, microprocessors, other electronic units designed toperform the functions described above, and/or a combination thereof. Thedisclosed OTFS methods will be described in view of the variouscomponents of the transceiver 400.

In one aspect a method of OTFS communication involves transmitting atleast one frame of data ([D]) from the transmitting device 310 to thereceiving device 330 through the communication channel 320, such frameof data comprising a matrix of up to N² data elements, N being greaterthan 1. The method comprises convolving, within the OTFS transceiver315-1, the data elements of the data frame so that the value of eachdata element, when transmitted, is spread over a plurality of wirelesswaveforms, each waveform having a characteristic frequency, and eachwaveform carrying the convolved results from a plurality of said dataelements from the data frame [D]. Further, during the transmissionprocess, cyclically shifting the frequency of this plurality of wirelesswaveforms over a plurality of times so that the value of each dataelement is transmitted as a plurality of cyclically frequency shiftedwaveforms sent over a plurality of times. At the receiving device 330,the OTFS transceiver 315-2 receives and deconvolves these wirelesswaveforms thereby reconstructing a replica of said at least one frame ofdata [D]. In the exemplary embodiment the convolution process is suchthat an arbitrary data element of an arbitrary frame of data ([D])cannot be guaranteed to be reconstructed with full accuracy untilsubstantially all of these wireless waveforms have been transmitted andreceived.

FIG. 5 illustrates a comparison of bit error rates (BER) predicted by asimulation of a TDMA system and an OTFS system. Both systems utilize a16 QAM constellation. The simulation modeled a Doppler spread of 100 Hzand a delay spread of 3 microsec. As can be seen from the graphs, theOTFS system offers much lower BER than the TDMA system for the samesignal-to-noise ratio (SNR).

Attention is now directed to FIG. 45, which is a flowchartrepresentative of the operations performed by an OTFS transceiver 4500which may be implemented as, for example, the OTFS transceiver 400. TheOTFS transceiver 4500 includes a transmitter including a modulator 4510and a receiver including a demodulator 4520 and two-dimensionalequalizer 4530. In operation, a transmitter of the OTFS transceiver 4500receives a two-dimensional symbol stream in the form of an N×N matrix ofsymbols, which may hereinafter be referred to as a TF matrix:x∈C ^(N×N)

As is illustrated in FIG. 46, in one embodiment the modulator 4510functions as an orthogonal map disposed to transform the two-dimensionalTF matrix to the following transmitted waveform:ϕ_(i) M(x)=Σx(i,j)ϕ_(i,j)ϕ_(i,j)⊥ϕ_(i,j)

Referring to FIG. 47, the demodulator 4520 transforms the receivedwaveform to a two-dimensional TF matrix in accordance with an orthogonalmap in order to generate an output stream:ϕ_(r)

y=D(ϕ_(r))

In one embodiment the OTFS transceiver 4500 may be characterized by anumber of variable parameters including, for example, delay resolution(i.e., digital time “tick” or clock increment), Doppler resolution,processing gain factor (block size) and orthonormal basis function. Eachof these variable parameters may be represented as follows.

Delay resolution (digital time tick):

${\Delta\; T} \in {R^{> 0}\left( {{\Delta\; T} = \frac{1}{Bw}} \right)}$

Doppler resolution:

${\Delta\; F} \in {R^{> 0}\left( {{\Delta\; F} = \frac{1}{Trans}} \right)}$

Processing gain factor (block size):N>0

Orthonormal basis of C^(N×1) (spectral shapes):U={u ₁ ,u ₂ , . . . ,u _(N)}

As is illustrated by FIG. 45, during operation the modulator 4510 takesa TF matrix x∈C^(N×N) and transforms it into a pulse waveform. In oneembodiment the pulse waveform comprises a pulse train defined in termsof the Heisenberg representation and the spectral shapes:

$\phi_{t} = {{M(x)} = \left( {\underset{\underset{b_{1}}{︸}}{{\Pi(x)}u_{1}},\underset{b_{2}}{\underset{︸}{{\Pi(x)}u_{2}}},\ldots\mspace{11mu},\underset{\underset{b_{N}}{︸}}{{\Pi(x)}u_{N}}} \right)}$

where b₁, b₂ . . . b_(N) are illustrated in FIG. 48 and where, inaccordance with the Heisenberg relation:Π(h*x)=Π(h)·Π(x) in particular:Π(δ_((t,0)) *x)=L _(τ)·Π(x)Π(δ_((0,w)) *x)=M _(x)·Π(x)

The Heisenberg representation provides that:

$\mspace{14mu}{\begin{matrix}{\Pi\text{:}\mspace{14mu}{C^{N \times N}\overset{\approx}{\longrightarrow}C^{N \times N}}\mspace{20mu}{g{iven}}{\mspace{11mu}\;}{by}\text{:}} & \; \\{{{\Pi(x)} = {\sum\limits_{\tau,{w = 0}}^{N - 1}{{x\left( {\tau,w} \right)}M_{w}L_{\tau}}}},{x \in C^{N \times N}}} & \;\end{matrix}\quad}$where L_(t) and M_(w) are respectively representative of cyclic time andfrequency shifts and may be represented as:

L_(τ) ∈ C^(N × N):  L_(τ)(φ)(t) = φ(t + τ), τ = 0, …  , N − 1${{M_{w} \in {C^{N \times N}\text{:}\mspace{14mu}{M_{w}(\varphi)}(t)}} = {e^{\frac{j\; 2\pi}{N}{wt}}{\varphi(t)}}},{w = 0},\ldots\mspace{14mu},{N - 1}$

The demodulator 4520 takes a received waveform and transforms it into aTF matrix y∈C^(N×N) defined in terms of the Wigner transform and thespectral shapes:

ϕ_(r) = (b₁, b₂, …  , b_(N))${y\left( {\tau,w} \right)} = {{{D\left( \phi_{r} \right)}\left( {\tau,w} \right)} = \overset{{Wigner}\mspace{14mu}{transform}}{\overset{︷}{\frac{1}{N}{\sum\limits_{n = 1}^{N}\left\langle {{M_{w}L_{\tau}u_{n}},b_{n}} \right\rangle}}}}$

Main property of M and D (Stone von Neumann theorem):D(h ^(a) M)x))=h*x where:h(τ,w)≈a(τΔT,wΔF)

As illustrated in FIG. 49, the equalizer 4530 may be implemented as atwo-dimensional decision feedback equalizer configured to perform aleast means square (LMS) equalization procedure such that:y

{circumflex over (x)}

Matrix Formulation

Throughout this description, the use of matrix terminology should beunderstood as being a concise description of the various operations thatwill be carried out by either the OTFS transceiver 315-1 or the OTFStransceiver 315-2. Thus the series of steps used to obtain thecoefficients of a particular matrix generally correspond to a set ofinstructions for the transmitter or receiver electronic circuitry (e.g.,the various components of the transmitter 405 and the receiver 455illustrated in FIG. 4A). For example, one set of coefficients mayinstruct the transmitter 405 or receiver 455 to perform a spreadspectrum operation, a different set of coefficients may instruct thetransmitter 405 or receiver 455 to perform a spectral shaping modulationor demodulation operation, and another set of coefficients may instructthe transmitter to perform various time spreading or time accumulationfunctions. Here standard matrix mathematics is used as a shorthand wayof reciting the series of instructions used to transmit and receivethese complex series of wireless signals.

Thus, when the discussion speaks of multiplying matrices, each dataelement in the matrix formed by the multiplication can be understood interms of various multi-step operations to be carried out by thetransmitter or receiver electronic circuitry (e.g., the transmitter 405or the receiver 455 as illustrated in FIG. 4A), rather than as a purenumber. Thus, for example, a matrix element formed from one matrix thatmay have spread-spectrum like pseudorandom numbers multiplied by anothermatrix that may have tone or spectral-shape spreading instructions, suchas QAM or phase shift keying instructions, multiplied by anotherscanning system, permutation scheme, or matrix that may have datainstructions, should be understood as directing the transmitter 405 totransmit a radio signal that is modulated according to these threemeans, or as directing the receiver 455 to receive and demodulate/decodea radio signal that is modulated according to these three means.

Put into matrix terminology, the OTFS method of convolving the data fora group of symbols over both time, spectrum, and tone or spectral-shapecan be viewed as transforming the data frame with N² informationelements (symbols) to another new matrix with N² elements whereby eachelement in the new transformed matrix, (here called the TFS data matrix)carries information about all elements of the original data frame. Inother words the new transformed TFS data matrix will generally carry aweighted contribution from each element of the original data framematrix [D]. Elements of this TFS data matrix are in turn transmitted andreceived over successive time intervals.

As previously discussed, in embodiments of the OTFS method the basicunit of convolution and deconvolution (convolution unit) is composed ofa matrix of N² symbols or data elements. Over each time interval, adifferent waveform may be used for each data element. By contrast, priorart methods generally use the same waveform for each data element. Forconsistency, the N² units of data will generally be referred to in thisspecification as a “data frame”. N may be any value greater than one,and in some embodiments will range from 64 to 256.

One distinction between the OTFS method and conventional modulationschemes may be appreciated by observing that a basic unit ofconvolution, transmission, reception and deconvolution for a prior artcommunications protocol may be characterized as a data frame of nsymbols or elements “d” operated on spreading codes that send the datafor n symbols over one spreading interval time where:[D _(1×n) ]=[d ₁ d ₂ . . . d _(n)]

In contrast, embodiments of the OTFS method generally use a differentbasic unit of convolution, transmission, reception, and deconvolution.In particular, such OTFS embodiments will typically use a larger dataframe [D_(N×N)] composed of N² elements or symbols “d” that, as will bediscussed, send the data for these N² elements over a plurality ofspreading interval times (often the plurality is N). The data frame[D_(N×N)] may be expressed as:

$\left\lbrack D_{N \times N} \right\rbrack = \begin{bmatrix}d_{1,1} & d_{1,2} & \ldots & d_{1,N} \\d_{2,1} & d_{2,2} & \ldots & d_{2,N} \\d_{3,1} & d_{3,2} & \ldots & \; \\d_{4,1} & d_{4,2} & \ldots & d_{N,N}\end{bmatrix}$

In general, references herein to a frame of data may be considered to bea reference to the N×N or N² matrix such as the one shown above, whereat least some elements in the matrix may be zero or null elements. Insome embodiments, a frame of data could be non-square, or N×M, whereN≠M.

Signal Transmission

As previously discussed, the OTFS method will spread this group of N²symbols across a communications link over multiple spreading intervalsof time (usually at least N spreading intervals or times), where eachindividual spreading interval of time is composed of at least N timeslices. Note that due to potential overhead for synchronization andidentification purposes, in some embodiments, extra time slices and/orextra spreading intervals of time may be allocated to provide room forthis overhead. Although for clarity of presentation this overhead willgenerally be ignored, it should be understood that the disclosure isintended to also encompass methods in which such overhead exists.

In exemplary embodiments of the OTFS method the data will thus betransmitted as a complex series of waveforms, usually over wirelessradio signals with frequencies usually above 100 MHz, and often above 1GHz or more. These radio frequencies are then usually received over atleast N spreading time intervals, where each spreading time interval isoften composed of at least N time-slices. Once received, the originaldata frame will be deconvolved (i.e. solved for) and the most likelycoefficients of the original group of symbols are reconstructed. Itshould be evident that in order to successfully deconvolve or solve forthe original data frame, the receiver will usually have prior knowledgeof the time, spectrum, and tone or spectral-shape spreading algorithmsused by the transmitter.

Attention is now directed to FIG. 3B, which illustrates a process 340 bywhich the OTFS transceiver 315-1 of the transmitting device 310 maytransmit a data frame (or a convolution unit) of data, here expressed asan (N by N) or (N²) matrix [D]. This process may be described usingstandard matrix multiplication as follows:

1: Construct the matrix product of a first N×N matrix [U1] and [D](often written as either [U₁]*[D] or more simply [U₁][D]—here both the“*” and simple close association (e.g. [U₁][D]) both are intended torepresent matrix multiplication) (stage 342).

2: Optionally permute [U₁][D] by a permutation operation P in order tocreate a new N×N matrix (stage 344). In general, any invertiblepermutation operation may be used. P may be an identity operation, oralternatively may be a permutation operation that essentially translatesthe columns of the original N×N [U₁][D] matrix to diagonal elements of atransformed [U₁][D]′ matrix.

3: Upon completing the permutation, optionally multiply the permutedresult by a second N×N [U₂] matrix (for spectral shaping for example),forming:

-   -   [P([U₁][D])][U₂] (stage 348).

4: Transmit this signal, according to methods discussed below (stage350).

In one embodiment the permutation operation P may optionally be of theform:b _(i,j) =a _(i,(j−i)mod N)where [a] is the original matrix (here [U₁][D]), and [b] is the newmatrix (here P([U₁][D]).

For sake of simplicity, the result of this permutation operation may bewritten as P([U₁] [D]).

FIG. 22 illustrates another permutation that may be used. In this case,the permutation is given by the following relationship:b _(i,j) =a _(i,(j+i)mod N)

Yet another permutation option is illustrated in FIG. 23. In FIG. 23,for illustrative purposes, a second [a] matrix is placed next to theoriginal [a] matrix. Diagonal lines are drawn overlapping the first andsecond [a] matrices. The permuted [b] matrix is formed by translatingeach diagonal line one column to the left (or right in yet anotherpermutation), where one or more of the translated entries falls into thesecond [a] matrix such that one or more entries is moved from the second[a] matrix to the same position in the first [a] matrix.

Here [U₁] and [U₂], if being used, can both be unitary N×N matrices,usually chosen to mitigate certain impairments on the (often wireless)communications link, such as wide band noise, narrow-band interference,impulse noise, Doppler shift, crosstalk, etc. To do this, rather thansimply selecting [U₁] and [U₂] to be relatively trivial identitymatrices [I], or matrices with most of the coefficients simply beingplaced along the central diagonal of the matrix, [U₁] and [U₂] willusually be chosen with non-zero coefficients generally throughout thematrix so as to accomplish the desired spreading or convolution of theconvolution unit [D] across spectrum and tone or spectral-shape space ina relatively efficient and uniform manner. Usually, the matrixcoefficients will also be chosen to maintain orthogonality or to providean ability to distinguish between the different encoding schemesembodied in the different rows of the respective matrices, as well as tominimize autocorrelation effects that can occur when radio signals aresubjected to multi-path effects.

In reference to the specific case where [U₁] may have rows thatcorrespond to pseudo-random sequences, it may be useful to employ apermutation scheme where each successive row in the matrix is acyclically rotated version of the pseudo-random sequence in the rowabove it. Thus the entire N×N matrix may consist of successivecyclically rotated versions of a single pseudo-random sequence of lengthN.

FIGS. 17-19 illustratively represent the manner in which different typesof unitary matrices [U₁] can be used to represent various forms ofmodulation. For example, FIG. 17 illustrates a unitary matrix [U1] inthe form of an identity matrix 1710 representative of a time divisionmultiplexed transmission basis; that is, a matrix of basis vectors whereeach column and each row is comprised of a single “1” and multiple “0”values. When the identity matrix 1710 is combined with a data matrix[D], the result corresponds to each column of [D] being transmitted in adifferent time slot corresponding to one of the time lines 1700 (i.e.,the columns of [D] are transmitted in a time division multiplexed seriesof transmissions).

FIG. 18 illustrates a unitary matrix [U₁] in the form of a DFT basisvector matrix 1810 representative of a frequency division multiplexedtransmission basis. The DFT basis vector matrix 1810 is comprised of Ncolumn entries representing rotating phaser or tone basis vectors. Whenthe DFT basis vector matrix 1810 is multiplied by a data matrix [D], thecolumns of the resulting matrix represent rotating phasers each having adifferent frequency offset or tone as represented by the set of timelines 1800. This corresponds to each column of [D] being transmitted ata different frequency offset or tone.

FIG. 19 illustrates a unitary matrix [U₁] in the form of a Hadamardmatrix 1910 representative of a code division multiplexed transmissionbasis. The Hadamard matrix 1910 is comprised of a set of quasi-randomplus and minus basis vectors. When the Hadamard matrix 1910 ismultiplied by a data matrix [D], the columns of the resulting matrixrepresent different quasi-random code division multiplexed signals asrepresented by the set of time lines 1900. This corresponds to eachcolumn of [D] being transmitted using a different quasi-random code.

In principle, [U₁] and [U₂], if both are being used, may be a widevariety of different unitary matrices. For example, [U₁] may be aDiscrete Fourier Transform (DFT) matrix and [U₂] may be a Hadamardmatrix. Alternatively [U₁] may be a DFT matrix and [U₂] may be a chirpmatrix. Alternatively [U₁] may be a DFT matrix and [U₂] may also be aDFT matrix, and so on. Thus although, for purposes of explaining certainaspects of the OTFS method, certain specific examples and embodiments of[U₁] and [U₂] will be given, these specific examples and embodiments arenot intended to be limiting.

Note that a chirp matrix, [V], is commonly defined in signal processingas a matrix where, if Ψ is the chirp rate,

[V]=diag(Ψ, Ψ², . . . Ψ^(n)), Ψ=e^(jψ), and frequency=e^(jω) where ω isthe initial center frequency of the spectrum.

Alternatively, a different chirp matrix may be used, filled withelements of the form:

$V_{j,k} = e^{(\frac{{- {i2}}\;\pi\;{kj}^{2}}{N})}$

Where j is the matrix row, k is the matrix column, and N is the size ofthe matrix.

Other commonly used orthonormal matrices, which may be used for [U₁], or[U₂] or [U₃] (to be discussed), include Discrete Fourier matrices,Polynomial exponent matrices, harmonic oscillatory, matrices, thepreviously discussed Hadamard matrices, Walsh matrices, Haar matrices,Paley matrices, Williamson matrices, M-sequence matrices, Legendrematrices, Jacobi matrices, Householder matrices, Rotation matrices, andPermutation matrices. The inverses of these matrices may also be used.

As will be discussed, in some embodiments, [U₁] can be understood asbeing a time-frequency shifting matrix, and [U₂] can be understood asbeing a spectral shaping matrix. In order to preserve readability, [U₁]will thus often be referred to as a first time-frequency shiftingmatrix, and [U₂] will thus often be referred to as a second spectralshaping matrix. However use of this nomenclature is also not intended tobe limiting. In embodiments in which the optional permutation ormultiplication by a second matrix [U2] is not performed, the [U1] matrixfacilitates time shifting by providing a framework through which theelements of the resulting transformed data matrix to be transmitted atdifferent times (e.g., on a column by column basis or any other orderedbasis).

Turning to some more specific embodiments, in some embodiments, [U1] mayhave rows that correspond to Legendre symbols, or spreading sequences,where each successive row in the matrix may be a cyclically shiftedversion of the Legendre symbols in the row above it. These Legendresymbols will occasionally also be referred to in the alternative as basevectors, and occasionally as spectrum-spreading codes.

In some embodiments, [U₂] may chosen to be a Discrete Fourier transform(DFT) matrix or an Inverse Discrete Fourier Transform matrix (IDFT).This DFT and IDFT matrix can be used to take a sequence of real orcomplex numbers, such as the N×N (P[U₁][D]) matrix, and further modulateP([U₁][D]) into a series of spectral shapes suitable for wirelesstransmission.

The individual rows for the DFT and IDFT matrix [U₂] will occasionallybe referred in the alternative as Fourier Vectors. In general, theFourier vectors may create complex sinusoidal waveforms (tone orspectral-shapes) of the type:

$X_{j}^{k} = e^{(\frac{{{- i^{*}}2^{*}\pi^{*}j^{*}k})}{N}}$where, for an N×N DFT matrix, X is the coefficient of the Fourier vectorin row k, column N of the DFT matrix, and j is the column number. Theproducts of this Fourier vector can be considered to be tones orspectral-shapes.

Although certain specific [U₁] and [U₂] can be used to transmit anygiven data frame [D], when multiple data frames [D] are beingtransmitted simultaneously, the specific [U₁] and [U₂] chosen may varybetween data frames [D], and indeed may be dynamically optimized toavoid certain communications link impairments over the course oftransmitting many data frames [D] over a communications session.

This process of convolution and modulation will normally be done by anelectronic device, such as a microprocessor equipped, digital signalprocessor equipped, or other electronic circuit that controls theconvolution and modulation parts of the wireless radio transmitter.Similarly the process of receiving and demodulation will also generallyrely upon a microprocessor equipped, digital signal processor equipped,or other electronic circuit that controls the demodulation,accumulation, and deconvolution parts of the wireless radio receiver.

Thus again using matrix multiplication, and again remembering that theseare all N×N matrices, [P([U₁][D])][U₂], where [U₂] is optional,represents the TFS data matrix that the transmitter will distribute overa plurality of time spreading intervals, time slices, frequencies, andspectral shapes. Note again that as a result of the various matrixoperation and optional permutation steps, a single element or symbolfrom the original N×N data matrix [D] after modulation and transmission,will be distributed throughout the different time spreading intervals,time slices, frequencies, and spectral shapes, and then reassembled bythe receiver and deconvolved back to the original single data element ofsymbol.

FIG. 6A illustratively represents an exemplary OTFS method 600 fortransmitting data over a wireless link such as the communication channel320. FIG. 6B illustrates components of an exemplary OTFS transmitter 650for performing the method of FIG. 6A. The method 600 can be performed,for example, by components of the OTFS transceiver 400 of FIG. 4 or bycomponents of the OTFS transmitter 650 of FIG. 6B.

In the example of FIG. 6, the payload intended for transmissioncomprises an input data frame 601 composed of an N×N matrix [D]containing N² symbols or data elements. As shown in FIG. 6A, asuccession of data frames 601 are provided, each of which defines amatrix [D] of N×N data elements. Each matrix [D] can be provided by adigital data source 660 in the OTFS transmitter 650. The elements of thematrix [D] can be complex values selected from points in a constellationmatrix such as, for example, a 16 point constellation of a 16QAMquantizer. In order to encode this data, an OTFS digital encoder 665will select an N×N matrix [U₁] 602 and, in some embodiments, select anN×N matrix [U₂] 604 (stage 606). As previously discussed, in someembodiments the matrix [U₁] 602 may be a matrix composed of Legendresymbols or a Hadamard matrix. This matrix [U₁] 602 will often bedesigned to time and frequency shift the symbols or elements in theunderlying data matrix [D] 601.

The matrix [U₂] 604 may be a DFT or IDFT matrix and is often designed tospectrally shape the signals. For example, in some embodiments thematrix [U₂] 604 may contain the coefficients to direct the transmittercircuits of the OTFS modulator 430 to transform the signals over time ina OFDM manner, such as by quadrature-amplitude modulation (QAM) orphase-shift keying, or other scheme.

Usually the matrix [D] 601 will be matrix multiplied by the matrix [U₁]602 by the digital encoder 665 at stage 610, and the matrix product ofthis operation [U₁][D] then optionally permuted by the digital encoder665 forming P([U₁][D]) (stage 611). In embodiments in which a spectralshaping matrix is utilized, the digital encoder 665 multiplies matrix[U₁][D] by matrix [U₂] 604 forming an N×N TFS data matrix, which mayalso be referred to herein as an OFTS transmission matrix (stage 614).

The various elements of the TFS matrix are then selected by the OTFSanalog modulator 670, usually a column of N elements at a time, on asingle element at a time basis (stage 616) The selected elements arethen used to generate a modulated signal that is transmitted via anantenna 680 (stage 618). More specifically, in one embodiment theparticular real and imaginary components of each individual TFS matrixelement are used to control a time variant radio signal 620 during eachtime slice. Thus one N-element column of the TFS matrix will usually besent during each time-spreading interval 608, with each element fromthis column being sent in one of N time slices 612 of the time-spreadinginterval 608. Neglecting overhead effects, generally a complete N×N TFSmatrix can be transmitted over N single time spreading intervals 622.

Attention is now directed to FIG. 6C, which is a flowchartrepresentative of an exemplary OTFS data transmission method 690 capableof being implemented by the OTFS transmitter 650 or, for example, by theOTFS transmitter 2100 of FIG. 21 (discussed below). As shown, the methodincludes establishing a time-frequency transformation matrix of at leasttwo dimensions (stage 692). The time-frequency transformation matrix isthen combined with a data matrix (stage 694). The method 690 furtherincludes providing a transformed matrix based upon the combining of thetime-frequency transformation matrix and the data matrix (stage 696). Amodulated signal is then generated in accordance with elements of thetransformed data matrix (stage 698).

Attention is now directed to FIG. 21A, which is a block diagramrepresentation of an OTFS transmitter module 2100 capable of performingfunctions of the OTFS transmitter 650 (FIG. 6B) in order to implementthe transmission method 600 of FIGS. 6A and 6C. With reference to FIG.21 and FIG. 6B, the transmitter 2100 includes a digital processor 2102configured for inclusion within the digital encoder 665 and a modulator2104 configured for inclusion within the analog modulator component 670.The digital processor 2102, which may be a microprocessor, digitalsignal processor, or other similar device, accepts as input the datamatrix [D] 2101 and may either generate or accept as inputs a [U₁]matrix 2102 and a [U₂] matrix 2104. A matrix generation routine 2105stored within a memory associated with the processor 2102 will, whenexecuted by the processor 2102, generate a TFS matrix 2108 (FIG. 21B),which will generally be comprised of a set of complex-valued elements.Once generated, scanning/selection routine 2106 will, when executed bythe processor 2102, select individual elements from the TFS matrix 2108matrix, often by first selecting one column of N elements from the TFSmatrix and then scanning down this column and selecting individualelements at a time. Generally one new element will be selected everytime slice 2112 (FIG. 21C).

Thus every successive time slice, one element from the TFS matrix 2108will be used to control the modulator 2104. In one embodiment of theOTFS method, the modulator 2104 includes modules 2132 and 2134 forseparating the element into its real and imaginary components, modules2142 and 2144 for chopping the resultant real and imaginary components,and filtering modules 2152 and 2154 for subsequently performingfiltering operations. The filtered results are then used to control theoperation of sin and cosine generators 2162 and 2164, the outputs ofwhich are upconverted using a RF carrier in order to produce an analogradio waveform 2120. This waveform then travels to the receiver where itis demodulated and deconvolved as will be described below with referenceto FIG. 7. Thus in this scheme (again neglecting overhead effects),element t_(1,1) from the first column of the TFS matrix can be sent inthe first time slice, and the Nth element from the first column of theTFS matrix can be sent in the last time slice of the first timespreading interval 2124. The next element t_(1,2) from the second columnof the TFS matrix can be sent in the first time slice of the second timespreading interval 2128, and so on. Thus, the modulator 2104 transmits acomposite waveform during each time spreading interval, where the valueof the waveform is determined by a different element of the TFS matrix2108 during each time slice of the spreading interval.

In an alternative embodiment, diagonals of the TFS data matrix may betransmitted over a series of single time-spreading intervals, onediagonal per single time-spreading interval, so that N diagonals of thefinal N×N transmission matrix are transmitted over N time intervals. Inother embodiments the order in which individual elements of the TFStransmission matrix [[U₁][D]][U₂] are transmitted across thecommunications link is determined by a transmit matrix or transmitvector.

In some embodiments, there may be some overhead to this basic model.Thus, for example, with some time padding (additional time slices oradditional time spreading intervals), checksums or otherverification/handshaking data, which could be transmitted in anon-convolved manner, could be sent back by the receiver to thetransmitter on a per time-spreading interval, per N time spreadingintervals, or even on a per time slice interval in order to requestretransmission of certain parts of the TFS data matrix as needed.

FIG. 9 illustratively represents an exemplary transmit frame 900comprised of a plurality of transmit blocks 920 separated by guard times950. Each transmit block 920 includes data corresponding to a portion ofthe [D] matrix, such as a column as shown in FIG. 9, or a row, orsub-blocks of the [D] matrix. The guard time 950 can provide thereceiver with time to resolve Doppler shift in transmitted signals. TheDoppler shift causes delays or advances in the receive time and the OTFSreceiver 455 can use the spaces between the transmit blocks 920-1,920-2, 920-3, 920-4 and 920-5 to capture data without interference fromother users. The guard times 950 can be used with either the first orsecond forms of the OTFS methodology. The guard times 950 can beutilized by other transmitters in the area so long as the transmissionuses different codes (e.g., Hadamard codes) than those used to transmitthe frame 900.

Attention is now directed to FIG. 20, which illustrates a sequence of LOTFS matrices 2010, each of dimension N×N. The L OFTS matrices 2010collectively comprise a frame of data which includes L×N×N symbolsspread out in both time and frequency. The matrices 2010-1 through2010-L are transmitted back-to-back and include guard times (T_(g))between matrices 2010. The N columns 2020 of a given matrix 2010 aretransmitted on a column-by-column basis, typically with guard timesinserted between the transmission of each column 2020. Therefore, the Lframes 2010 are transmitted in a time greater than N×[L×(N×T+Tg)], whereT is the time to transmit one column of symbols inclusive of the guardtimes described above.

As previously discussed, in some embodiments, the first N×N timespreading matrix [U₁] may be constructed out of N rows of a cyclicallyshifted Legendre symbols or pseudorandom number of length N. That is,the entire N×N spreading matrix is filled with all of the various cyclicpermutations of the same Legendre symbols. In some embodiments, thisversion of the [U₁] matrix can be used for spectrum spreading purposesand may, for example, instruct the transmitter to rapidly modulate theelements of any matrix that it affects rapidly over time, that is, witha chip rate that is much faster than the information signal bit rate ofthe elements of the matrix that the Legendre symbols are operating on.

In some embodiments, the second N×N spectral shaping matrix [U₂] can bea Discrete Fourier Transform (DFT) or an Inverse Discrete FourierTransform (IDFT) matrix. These DFT and IDFT matrices can instruct thetransmitter to spectrally shift the elements of any matrix that the DFTmatrix coefficients act upon. Although many different modulation schemesmay be used, in some embodiments, this modulation may be chosen to be anOrthogonal Frequency-Division Multiplexing (OFDM) type modulation, inwhich case a modulation scheme such as quadrature amplitude modulationor phase-shift keying may be used, and this in turn may optionally bedivided over many closely-spaced orthogonal sub-carriers.

Often the actual choice of which coefficients to use for the first N×Ntime-frequency shifting matrix [U₁] and what coefficients to use for thesecond N×N spectral shaping matrix [U₂] may depend on the conditionspresent in the communications channel 320. If, for example, acommunications channel 320 is subjected to a particular type ofimpairment, such as wide band noise, narrow-band interference, impulsenoise, Doppler shift, crosstalk and so on, then some first N×Ntime-frequency shifting matrices and some second N×N spectral shapingmatrices will be better able to cope with these impairments. In someembodiments of the OTFS method, the transmitter and receiver willattempt to measure these channel impairments, and may suggest alternatetypes of first N×N time-frequency shifting matrices [U₁] and second N×Nspectral shaping matrices to each [U₂] in order to minimize the dataloss caused by such impairments.

Various modifications of the above-described data transmission processrepresented by the matrix multiplication [[U₁][D]][U₂] are also withinthe scope of the present disclosure and are described below withreference to FIGS. 13 and 15. For example, FIGS. 13A and 13B show afirst alternative OTFS transmission scheme. In the embodiment of FIG.13, the data matrix [D] may be further convolved by means of a thirdunitary matrix [U₃] 1306, which may be an IDFT matrix. In oneimplementation [U1] may be a DFT matrix and the matrix [U₂] 1308 may bethe product of a DFT matrix and a base. In this scheme, the process ofscanning and transmitting the data is represented by the previouslydescribed permutation operation P. The basic transmission process canthus be represented as [U₃]*[P([U₁][D])]*[U₂]. FIG. 13A shows the matrix[D] identified by reference numeral 1300, and the matrix product([U₁][D]) is identified by reference numeral 1302. FIG. 13A furthershows that the result of the permutation operation P is the permutedversion of the matrix product ([U₁][D]), i.e., P([U₁][D]), identified byreference numeral 1304. In the representation of FIG. 13A, at leastcertain of the effects of the permutation operation P are represented bythe differing directions of arrow 1305 and arrow 1305′.

FIG. 13B shows a result 1310 of the final matrix product[U₃][P([U₁][D])][U₂] where, again, the permuted version of the matrixproduct ([U₁][D]), i.e., P([U₁][D]), is identified by reference numeral1304 (without arrow 1305′). In various embodiment the matrix [U₃] 1306may comprise a DFT matrix, an IDFT matrix, or a trivial identity matrix(in which case this first alternative scheme becomes essentialequivalent to a scheme in which a matrix [U₃] is not employed).

Attention is now directed to FIGS. 15A and 15B, which illustrate asecond alternative OTFS transmission scheme. As shown, the original datamatrix [D] is identified by reference numeral 1500, the matrix product[U₁][D] is identified by reference numeral 1502, the permuted matrixP([U₁][D]) is identified by reference numeral 1504, and the matrix [U₂]is identified by reference numeral 1506. In the representation of FIG.15A, at least certain of the effects of the permutation operation P arerepresented by the differing directions of arrow 1507 and arrow 1507′.In one embodiment [U₁] may be a Hadamard matrix; that is, a squarematrix composed of mutually orthogonal rows and either +1 or −1coefficients. This matrix has the property that H*H^(T)=nI_(n) whereI_(n) is an N×N identity matrix and H^(T) is the transpose of H). Asshown in FIG. 15B, consistent with this alternative OTFS transmissionscheme the matrix corresponding to the transmitted signal may beexpressed as [P([U₁][D])]*[U₂] and is identified by reference numeral1508 where, again, the permuted matrix P([U₁][D]) is identified byreference numeral 1504 (without arrow 1507′).

Signal Reception and Data Reconstruction

Attention is now directed to FIG. 3C, which illustrates a process 360 bywhich the OTFS transceiver 315-2 of the receiving device 330 may operateto receive a transmitted data frame. Within the OTFS transceiver 315-2,the process performed during transmission is essentially done inreverse. Here the time and frequency spread replica of the TFS datamatrix ([P([U₁][D])][U₂])′ (where the ′ annotation is indicative of areplicated matrix) is accumulated over multiple time spreadingintervals, time slices, frequencies, and spectral shapes, and thendeconvolved to solve for [D] by performing the following operations:

1: Receive ([P([U₁][D])][U₂])′ (stage 362)

2: Perform a first left multiplication by the Hermitian matrix of the[U2] matrix [U₂ ^(H)], if one was used for transmission, thus creatingP([U₁][D]) (stage 364).

3: Inverse permute this replica by (P([U₁][D])P⁻¹, if a permutation wasused during transmission, thus creating [U₁][D] (stage 368)

4. Perform a second right multiplication by the Hermitian matrix of the[U1] matrix [U₁ ^(H)], thus recreating [D] (stage 370).

As a consequence of noise and other impairments in the channel, use ofinformation matrices and other noise reduction methods may be employedto compensate for data loss or distortion due to various impairments inthe communications link. Indeed, it may be readily appreciated that oneadvantage of spreading out the original elements of the data frame [D]over a large range of times, frequencies, and spectral shapes ascontemplated by embodiments of the OTFS method is that it becomesstraightforward to compensate for the loss during transmission ofinformation associated with a few of the many transmission times,frequencies and spectral shapes.

Although various deconvolution methods may used in embodiments of theOTFS method, the use of Hermitian matrices may be particularly suitablesince, in general, for any Hermitian matrix [U^(H)] of a unitary matrix[U], the following relationship applies:

[U][U^(H)]=[I] where [I] is the identity matrix.

Communications links are not, of course, capable of transmitting data atan infinite rate. Accordingly, in one embodiment of the OTFS method thefirst N×N time-frequency shifting matrix ([U₁], the second N×N spectralshaping matrix ([U₂] (when one is used), and the elements of the dataframe, as well as the constraints of the communications link (e.g.available bandwidth, power, amount of time, etc.), are chosen so that inbalance (and neglecting overhead), at least N elements of the N×N TFSdata matrix can be transmitted over the communications link in onetime-spreading interval. More specifically (and again neglectingoverhead), one element of the N×N TFS data matrix will generally betransmitted during each time slice of each time-spreading interval.

Given this rate of communicating data, then typically the entire TFSdata matrix may be communicated over N time-spreading intervals, andthis assumption will generally be used for this discussion. However itshould be evident that given other balancing considerations between thefirst N×N time-frequency shifting matrix, the second N×N spectralshaping matrix, and the elements of the data frame, as well as theconstraints of the communications link, the entire TFS data matrix maybe communicated in less than N time-spreading intervals, or greater thanN time spreading intervals as well.

As discussed above, the contents of the TFS data matrix may betransmitted by selecting different elements from the TFS data matrix,and sending them over the communications link, on a one element per timeslice basis, over multiple spreading time intervals. Although inprinciple this process of selecting different elements of the TFS datamatrix can be accomplished by a variety of different methods, such assending successive rows of the TFS data matrix each single timespreading interval, sending successive columns of the TFS data matrixeach successive time spreading interval, sending successive diagonals ofthe TFS data matrix each successive time spreading intervals, and so on,from the standpoint of communications link capacity, minimizinginterference, and reducing ambiguity, some schemes are often better thanothers. Thus, often the [U₁] and [U₂] matrices, as well as thepermutation scheme P, may be chosen to optimize transmission efficiencyin response to various impairments in the communications link.

As shown in FIG. 4B, an exemplary process 404 pursuant to which an OTFStransceiver may transmit, receive and reconstruct information utilizinga TFS data matrix may thus be generally characterized as follows:

1: For each single time-spreading interval, selecting N differentelements of the TFS data matrix (often successive columns of the TFSmatrix will be chosen) (stage 482).

2: Over different time slices in the given time spreading interval,selecting one element (a different element each time slice) from the Ndifferent elements of the TFS data matrix, modulating this element, andtransmitting this element so that each different element occupies itsown time slice (stage 484).

3: Receiving these N different replica elements of the transmitted TFSdata matrix over different said time slices in the given time spreadinginterval (stage 486).

4: Demodulating these N different elements of the TFS data matrix (stage488).

5. Repeating stages 482, 484, 486 and 488 up to a total of N times inorder to reassemble a replica of the TFS data matrix at the receiver(stage 490).

This method assumes knowledge by the receiver of the first N×N spreadingcode matrix [U₁], the second N×N spectral shaping matrix [U₂], thepermutation scheme P, as well as the particular scheme used to selectelements from the TFS matrix to transmit over various periods of time.In one embodiment the receiver takes the accumulated TFS data matrix andsolves for the original N×N data frame using standard linear algebramethods. It may be appreciated that because each original data symbolfrom the original data frame [D] has essentially been distributed overthe entire TFS data matrix, it may not be possible to reconstruct anarbitrary element or symbol from the data [D] until the complete TFSdata matrix is received by the receiver.

Attention is now directed to FIG. 7A, which illustratively represents anexemplary method 700 for demodulating OTFS-modulated data over awireless link such as the communication channel 320. FIG. 7B illustratescomponents of an exemplary OTFS receiver for performing the method ofFIG. 7A. The method 700 can be performed by the OTFS receiver module 455of the OTFS transceiver 400 of FIG. 4A or by the OTFS receiver 750 ofFIG. 7B. Just as the OTFS transmitter 405 is often a hybridanalog/digital device, capable of performing matrix calculations in thedigital portion and then converting the results to analog signals in theanalog portion, so to the OTFS receiver 750 will typically be capable ofreceiving and demodulating the radio signals in the analog receiver 770of the OTFS receiver 750, and then often decoding or deconvolving thesesignals in the digital portion of the digital OTFS receiver 780.

As shown in FIG. 7A, received signals 720 corresponding tochannel-impaired versions of the transmitted radio signals 620 may bereceived by, for example, an antenna 760 of the OTFS receiver 750. Thereceived signals 720 will generally not comprise exact copies of thetransmitted signals 620 because of the signal artifacts, impairments, ordistortions engendered by the communication channel 320. Thus replicas,but not exact copies, of the original elements of the TFS matrix arereceived and demodulated 722 by the OTFS analog receiver 770 every timeslice 612. In an exemplary embodiment one column of the TFS matrix isdemodulated at stage 722 during every spreading time interval 608. As aconsequence, the OTFS demodulator 460 will accumulate these elementsover N single time spreading intervals, eventually accumulating theelements necessary to create a replica of the original TFS matrix (stage724)

In order to decode or deconvolve the TFS matrix accumulated during stage724, the digital OTFS data receiver 780 left multiplies, during a stage726, the TFS matrix by the Hermitian matrix of the [U₂] matrix, i.e.,[U₂ ^(H)], established at a stage 704. Next, the digital OTFS datareceiver 780 performs, at a stage 728, an inverse permutation (P⁻¹) ofthe result of this left multiplication. The digital OTFS data receiver780 then deconvolves the TFS matrix in order to reconstruct a replica732 of the original data matrix [D] by, in a stage 730, rightmultiplying the result of stage 728 by the Hermitian of the original N×Ntime-frequency shifting matrix [U₁], i.e., [U₁ ^(H)], established at astage 702. Because the reconstructed signal will generally have somenoise and distortion due to various communications link impairments,various standard noise reduction and statistical averaging techniques,such as information matrices, may be used to assist in thereconstruction process (not shown). Each replicated frame 732 of eachoriginal data matrix [D] may be stored within digital data storage 782(stage 740).

Attention is now directed to FIG. 7C, which is a flowchartrepresentative of an exemplary OTFS data demodulation method 790 capableof being implemented by the OTFS receiver module 455 of the OTFStransceiver 400 or, for example, by the OTFS receiver 750 of FIG. 7B. Asshown in FIG. 7C, the method includes establishing a time-frequencydetransformation matrix of at least two dimensions (stage 792). Themethod further includes receiving a modulated signal formed using atime-frequency transformation matrix that is a hermetian of thedetransformation matrix (stage 794). The modulated signal is thendemodulated to form a transformed data matrix (stage 796). The methodfurther includes generating a data matrix by combining the transformeddata matrix and the detransformation matrix (stage 798).

Attention is now directed to FIGS. 16A and 16B, which illustrate analternative OTFS signal reception scheme corresponding to thealternative OTFS transmission scheme of FIG. 15. As shown in FIG. 16A,the matrix [r] 1600 of received data is demodulated and deconvolved(decoded) by forming the Hermitian matrices of the matrices [U₁] and[U₂] used to encode and modulate the data [D], as well as the inversepermutation operation P⁻¹ to undo the original permutation operation Pused to scan and transmit the data over multiple time intervals. In theillustration of FIGS. 16A and 16B, the inverse permutation P⁻¹([r][U₂^(H)]) is identified by reference numeral 1604 and the reconstructeddata matrix [D] (created from [U₁ ^(H)]*P⁻¹([r]*[U₂ ^(H)])) isidentified by the reference numeral 1606.

Attention is now directed to FIG. 15, which illustrates an alternativeOTFS transmission scheme. As shown, the original data matrix [D] isidentified by reference numeral 1500, the matrix product [U₁][D] isidentified by reference numeral 1502, the permuted matrix P([U₁][D]) isidentified by reference numeral 1504, and the matrix [U₂] is identifiedby reference numeral 1506. In the representation of FIG. 15, at leastcertain of the effects of the permutation operation P are represented bythe differing directions of arrow 1507 and arrow 1507′. In oneembodiment [U₁] may be a Hadamard matrix; that is, a square matrixcomposed of mutually orthogonal rows and either +1 or −1 coefficients.This matrix has the property that H*H^(T)=nI_(n) where I_(n) is an N×Nidentity matrix and H^(T) is the transpose of H). Consistent with thealternative OTFS transmission scheme of FIG. 15, the matrixcorresponding to the transmitted signal may be expressed as[P([U₁][D])]*[U₂] and is identified by reference numeral 1508.

Various modifications of the above-described data reconstruction processare also within the scope of the present disclosure and are describedbelow with reference to FIGS. 14 and 16. Turning now to FIGS. 14A and14B, there is illustrated a scheme for reception and reconstruction ofsignals transmitted consistent with the first alternative OTFStransmission scheme of FIG. 13. In FIG. 14A, the data that thetransmitter has received and accumulated, after various communicationslink impairment effects, is represented as the [r] matrix 1400. The [r]matrix 1400 is demodulated and deconvolved (decoded) by forming theHermitian matrices of the original [U₁], [U₂], and [U₃] matricesoriginally used to encode and modulate the data [D], as well as theinverse permutation operation P⁻¹ to undo the original permutationoperation P used to scan and transmit the data over multiple timeintervals. Here [U₁ ^(H)] may be an IDFT matrix, [U₃ ^(H)] may be a DFTmatrix, and [U₂ ^(H)] 1402 may be a DFT matrix times a base. As shown inFIG. 14A, P⁻¹([r][U₂ ^(H)]) is identified by the reference numeral 1403.In FIG. 14B, P⁻¹([U₃ ^(H)][r][U₂ ^(H)]) is identified by the referencenumeral 1404 and the reconstructed data matrix [D] is identified byreference numeral 1406.

Referring now to FIG. 11, there is illustrated an exemplary receiveframe 1100 including guard times 1150 between groups of received data orblocks 1120. The receive frame 1100 corresponds to a frame received inresponse to transmission of a frame having characteristics equivalent tothose illustrated in FIG. 9. As shown in FIG. 11, each receive block1120 includes information comprising a portion of the [D] matrix, suchas a column as shown in FIG. 11, or a row, or sub-blocks of the [D]matrix. The entire [D] matrix is received in a time T_(f) 1130 thatincludes N blocks 1120 and N-l guard times 1150. The guard time 1150provides the receiver with time to resolve Doppler shift in the receivedsignals. The Doppler shift causes delays or advances in the receive timeand the OTFS receiver 455 can use the guard times 1120 between thereceive blocks 1120-1, 1120-2, 1120-3, 1120-4 and 1120-5 to capture datawithout interference from other users.

Second Form of OTFS Method

Attention is now directed to FIGS. 8, 10 and 12, to which reference willbe made in describing aspects of a second form of the OTFS method. Asmentioned previously, in the first OTFS method, which was described withreference to FIGS. 6 and 7, data is transmitted on a per time slicebasis. In contrast, the second form of the OTFS method contemplates thatdata is transmitted as a series of waveforms, each of which generallysubsists for a period of N time slices. More particularly, inembodiments of the second form of the OTFS method each data elementwithin an input frame of data [D] including N² data elements is assigneda unique waveform derived from a basic waveform of duration N timeslices. In one implementation this uniqueness is obtained by assigningto each data element a specific combination of a time and frequencycyclic shift of the basic waveform.

Consistent with one embodiment of the second form of the OTFS method,each element in the input frame of data [D] is multiplied by itscorresponding unique waveform, thereby producing a series of N² weightedunique waveforms. Over one spreading time interval (generally composedof N time slices), all N² weighted unique waveforms corresponding toeach data element in the fame of data [D] are simultaneously combinedand transmitted. Further, in this embodiment a different unique basicwaveform of length (or duration) of N time slices is used for eachconsecutive time-spreading interval. The set of N unique basic waveforms(i.e., one for each of the N time-spreading intervals) form anorthonormal basis. As may be appreciated, embodiments of the second formof the OTFS element contemplate that at least a part of [D] istransmitted within each of the N time-spreading intervals.

To receive waveforms modulated and transmitted in accordance with thissecond form of the OTFS method, the received signal is (over eachspreading interval of N time slices), correlated with the set of all N²waveforms previously assigned to each data element during thetransmission process for that specific time spreading interval. Uponperforming this correlation, the receiver will produce a uniquecorrelation score for each one of the N² data elements (the receiverwill have or be provided with knowledge of the set of N² waveformsrespectively assigned by the transmitter to the corresponding set of N²data elements). This process will generally be repeated over all Ntime-spreading intervals. The original data matrix [D] can thus bereconstructed by the receiver by, for each data element, summing thecorrelation scores over N time-spreading intervals. This summation ofcorrelation scores will typically yield the N² data elements of theframe of data [D].

Turning now to FIG. 8, there are shown an exemplary set of vectors usedin convolving and deconvolving data in accordance with the second formof the OTFS method. Specifically, FIG. 8 depicts a base vector 802, datavector 800, Fourier vector 804 and Transmit vector 806. In theembodiment of FIG. 8 the data vector 800 may include N elements (oftenone row, column, or diagonal) of an N×N [D] matrix, the base vector 802may include N elements (often one row, column, or diagonal) of an N×N[U₁] matrix, the Fourier vector 804 may include N elements (often onerow, column, or diagonal) of an N×N [U₂] matrix, which may oftencomprise a DFT or IDFT matrix. The transmit frame 808 is composed of Nsingle time-spreading intervals T^(m) 810, each of which is defined by atransmit vector 806 containing multiple (such as N) time slices. In theembodiment of FIG. 8, the transmit vector 806 provides information usedby the transmitter in selecting elements of the OTFS transmission matrixfor transmission during each time slice of each transmission interval.

In FIG. 8, the lines 812 are intended to indicate that each Fouriervector waveform 804 is manifested over one spreading time interval T^(m)810. It is observed that this is representative of a difference inwireless radio signal modulation between the second form of the OTFSmethod (in which each waveform exists over a time spreading intervalcomposed of multiple (e.g. N) time slices) and the first form of theOTFS method (in which the wireless signal is essentially transmitted ona per time slice basis).

FIG. 10 illustrates aspects of a cyclic convolution method that may beused to convolve data and transmit data according to the second form ofthe OTFS methodology. As previously discussed, particularly in the casewhere [U₁] is composed of a cyclically permuted Legendre number oflength N, the process of convolving the data and scanning the data canbe understood alternatively as being a cyclic convolution of theunderlying data. Here the d⁰, d^(k), d^(N−1) can be understood as beingthe elements or symbols of the data vector 1000 component of the [D]matrix, the b^(m) coefficients can be understood as representing thebase vector 1002 components of the [U₁] matrix, and the X coefficientscan be understood as representing the Fourier vector 1004 components ofthe [U₂] matrix. The combinations of the b^(m) coefficients and the Xcoefficients are summed to form the transmit block T^(m) 1010. In theillustration of FIG. 10, each such combination is represented as[b^(m)*X^(k)] and comprises the element-wise multiplication of them^(th) base vector with the k^(th) Fourier vector.

FIGS. 39A, 39B, 39C and 39D illustrate an exemplary OTFS encoding schemepursuant to which N² data symbols d_(ij) of a data matrix are spreadusing a pair of transform matrices into N² different basis matricesB_(ij) of basis frames F_(ij). With reference to FIG. 39A, a basismatrix includes N length N basis vectors b₀-b_(N−1). When [U1] isimplemented using a DFT or IDFT matrix, the multiplication of the [D]matrix by [U1] and [U2] can be replicated by multiplying each of thebasis vectors b₀-b_(N−1) by a diagonal matrix formed by placing the Ncomponents of each DFT vector (column) along the main diagonal. Theresult of these multiplications is N² basis matrices. As shown in FIG.39A, each data element d_(ij) is then multiplied by one of the N² basismatrices and the resulting N² matrices d_(ij)*B_(ij) are summed to yieldan OTFS data matrix. This is illustrated by, for example, the cyclicconvolution of FIG. 10. Thus, each data element d_(ij) is spread overeach element of the OTFS data matrix.

FIG. 39B illustrates an incomplete basis matrix that includes N-lcolumns and N-k rows where l and k are greater than or equal to one. Theresulting multiplications spread only a portion of the data elementsd_(ij) over the entire N×N OTFS matrix. FIG. 39C illustrates a basisframe that has N vectors of length M where M is greater than N. Theresulting basis frames include N×M elements. FIG. 39D illustrates anincomplete basis frame including N-l columns and M-k rows, where l and kare greater than or equal to one. The result is that fewer than all thedata elements d_(ij) are spread across all of the N² basis frames.

FIG. 12 shows a diagram of a cyclic deconvolution method that may beused to deconvolve received data according to the second form of theOTFS methodology. In FIG. 12, R^(m) 1202 denotes a portion of theaccumulated signal 730 received and demodulated by the OTFS receiver455. Again, as previously discussed, particularly in the case where [U1]is composed of a cyclically permuted Legendre number of length N, thenthe matrix-based mathematical process of deconvolving the data andreconstructing the data can be understood alternatively as being acyclic deconvolution of the transmitted data previously convolved inFIG. 10. Here the reconstructed components 1200 ˜d⁰, ˜d^(k), ˜d^(N−1)can be understood as being the reconstructed elements (symbols) of thedata vector 1000 component of the [D] matrix, the b^(m) coefficients1002 again can be understood as representing the same base vector 1002components of the [U1] matrix, and the X coefficients 1004 can again beunderstood as representing the Fourier vector 1004 components of the[U2] matrix. In addition, [b^(m)*X^(k)]′ may be understood as denotingthe element-wise multiplication of the mirror conjugate of the m^(th)base vector with the k^(th) Fourier vector.

In this alternative scheme or embodiment, the OTFS method can beunderstood as being a method of transmitting at least one frame of data([D]) over a communications link, comprising: creating a plurality oftime-spectrum-tone or spectral-shape spreading codes operating over aplurality of time-spreading intervals, each single time-spreadinginterval being composed of at least one clock intervals; eachtime-spectrum-tone or spectral-shape spreading code comprising afunction of a first time-frequency shifting, a second spectral shaping,and a time spreading code or scanning and transmission scheme.

Multiple Users

In an exemplary embodiment, OTFS modulation techniques may be employedto enable data to be sent from multiple users using multipletransmitters (here usually referred to as the multiple transmitter case)to be received by a single receiver. For example, assume multiple users“a”, “b”, “c”, and “d”, each desire to send a frame of data including Nelements. Consistent with an embodiment of a multi-user OTFStransmission scheme, a conceptual N×N OTFS transmission matrix shared bymultiple users may be created in the manner described below.Specifically, each given user packs their N elements of data into onecolumn of an N×N data frame associated with such user but leaves theother columns empty (coefficients set to zero). The N×N data frame[D_(a)] associated with, and transmitted by, a user “a” may thus berepresented as:

$\left\lbrack D_{a} \right\rbrack = \begin{bmatrix}a_{1,1} & 0_{1,2} & \ldots & 0_{1,n} \\a_{2,1} & 0_{2,2} & \ldots & 0_{2,n} \\\ldots & \ldots & \ldots & \ldots \\a_{n,1} & 0_{n,2} & \ldots & 0_{n,n}\end{bmatrix}$

Similarly, the N×N data frame [Db] associated with, and transmitted by,a user “b” may thus be represented as

$\left\lbrack D_{b} \right\rbrack = \begin{bmatrix}0_{1,1} & b_{1,2} & \ldots & 0_{1,n} \\0_{2,1} & b_{2,2} & \ldots & 0_{2,n} \\\ldots & \ldots & \ldots & \ldots \\0_{n,1} & b_{n,2} & \ldots & 0_{n,n}\end{bmatrix}$

And user “n” sends and N×N data frame [D_(n)]

$\left\lbrack D_{n} \right\rbrack = \begin{bmatrix}0_{1,1} & 0_{1,2} & \ldots & n_{1,n} \\0_{2,1} & 0_{2,2} & \ldots & n_{2,n} \\\ldots & \ldots & \ldots & \ldots \\0_{n,1} & 0_{n,2} & \ldots & m_{n,n}\end{bmatrix}$

Thus, transmission of the data frames [D_(a)], [D_(b)] . . . [D_(n)]respectively by the users “a”, “b” . . . “n” results in transmission ofthe conceptual N×N OTFS transmission matrix, with each of the usersbeing associated with one of the columns of such conceptual transmissionmatrix. In this way each independent user “a”, “b” . . . “n” transmitsits N data elements during its designated slot (i.e., column) within theconceptual N×N OTFS transmission matrix, and otherwise does not transmitinformation. This enables signals corresponding to the data frames[D_(a)], [D_(b)] . . . [D_(n)] to be received at the receiver as if theconceptual N×N OTFS transmission matrix was representative of a completedata frame sent by only a single transmitter. Once so received at thereceiver, the received data frames [D_(a)], [D_(b)] . . . [D_(n)]effectively replicate the conceptual N×N OTFS transmission matrix, whichmay then be deconvolved in the manner discussed above.

FIG. 24 depicts a time/frequency plane 2400 which illustrates the mannerin which multiple users may transmit data in designated columns of aconceptual OTFS transmission matrix consistent with the precedingexample. As shown, the time/frequency plane 2400 includes a first tileT⁰ 2410-1 representative of transmission, by a first user, of data in afirst column of the conceptual OTFS transmission matrix. In theembodiment of FIG. 24 the first tile T⁰ 2410-1 encompasses an entirebandwidth (BW) of the OTFS channel and extends for a duration ofT_(f)/N, where T_(f) denotes a total time required to transmit all ofthe entries within the conceptual OTFS transmission matrix. Similarly,the time/frequency plane 2400 includes a second tile T¹ 2410-2representative of transmission, by a second user, of data in a secondcolumn of the conceptual OTFS matrix during a second T_(f)/N interval.In this way each of the N users are provided with a time interval ofT_(f)/N to transmit their respective N elements included within the N×Nconceptual OTFS transmission matrix.

FIG. 25 depicts an alternative time/frequency plane 2400 whichillustrates another manner in which multiple users may transmit data indesignated rows of a conceptual OTFS transmission matrix consistent withthe preceding example. As shown, the time/frequency plane 2500 includesa first tile T⁰ 2510-1 representative of transmission, by a first user,of data in a first row or first set of rows of the conceptual OTFStransmission matrix. In the embodiment of FIG. 25 the first tile T⁰2510-1 encompasses a first portion of the entire bandwidth (BW) of theOTFS channel corresponding to the number of first rows, and thetransmission extends for an entire duration T_(f), where T_(f) denotes atotal time required to transmit all of the entries within the conceptualOTFS transmission matrix. Similarly, the time/frequency plane 2500includes a second tile T¹ 2510-2 representative of transmission, by asecond user, of data in a second row or rows of the conceptual OTFSmatrix encompassing a second portion of the bandwidth, and alsotransmitting during the entire T_(f) time interval. In this way each ofthe users are provided with a portion of the bandwidth for the entiretime interval of T_(f) to transmit their respective N elements (orinteger multiple of N elements) included within the N×N conceptual OTFStransmission matrix.

FIG. 26 depicts yet another time/frequency plane 2600 which illustratesanother manner in which multiple users may transmit data in designatedcolumn/row portions of a conceptual OTFS transmission matrix consistentwith the preceding example. As shown, the time/frequency plane 2600includes a first tile T⁰ 2610-1 representative of transmission, by afirst user, of data in one or more first columns and one or more firstrows of the conceptual OTFS transmission matrix. In the embodiment ofFIG. 26 the first tile T⁰ 2610-1 encompasses a portion of the entirebandwidth (BW) of the OTFS channel proportional to the number of rows inthe first tile 2610-1, and the transmission extends for a duration of nT_(f)/N, where T_(f) denotes a total time required to transmit all ofthe entries within the conceptual OTFS transmission matrix and n≤Nrepresents the number of rows that the first tile 2610-1 includes.Similarly, the time/frequency plane 2600 includes a second tile T¹2610-2 representative of transmission, by a second user, of data in aone or more second columns and one or more second rows of the conceptualOTFS matrix during a second m T_(f)/N interval, where m≤N represents thenumber of rows in the second tile 2610-2. In this way each of the usersare provided with a time interval of an integer multiple of T_(f)/N totransmit their respective elements included within the N×N conceptualOTFS transmission matrix.

The size of the tiles in FIGS. 24-26 corresponds proportionally to theamount of data provided to the corresponding user. Therefore, users withhigher data rate requirements can be afforded larger portions of the [D]matrix and therefore larger tiles. In addition, users that are closer tothe transmitter can be afforded larger portions of the [D] matrix whileusers further away may be provided smaller portions to take advantage ofthe efficient transmissions to close users and minimize data losttransmitting to further users.

Multiple users that are using different transmitters (or simply multipletransmitters) may communicate over the same communications link usingthe same protocol. Here, each user or transmitter may, for example,select only a small number of data elements in the N² sized frame ofdata to send or receive their respective data. As one example, a usermay simply select one column of the frame of data for their purposes,and set the other columns at zero. The user's device will then computeTFS data matrices and send and receive them as usual.

As previously discussed, one advantage of the OTFS approach is increasedresistance to Doppler shifts and frequency shifts. For example, in manycases the greater degree of time, frequency, and spectral shapingcontemplated by the OTFS approach will largely mitigate any negativeeffects of such shifts due to the superior ability of OTFS-equippeddevices to function over an impaired communications link. In othercases, because the local impaired device can be identified with greateraccuracy, a base station or other transmitting device can either sendcorrective signals to the impaired device, or alternatively shut off theimpaired device.

Improving Resistance to Channel Impairments

As previously discussed, one advantage of the OTFS method is increasedresistance to communications channel impairments. This resistance toimpairments can be improved by further selecting the first N×Ntime-frequency shifting matrix and the second N×N spectral shapingmatrix to minimize the impact of an aberrant transmitter; specifically,a transmitter suffering from Doppler shift or frequency shift on theelements of the TFS data matrix that are adjacent to the elements of theTFS data matrix occupied by the aberrant transmitter. Alternatively, thereceiver may analyze the problem, determine if an alternate set of firstN×N time-frequency shifting matrices and/or said second N×N spectralshaping matrices would reduce the problem, and suggest or command thatcorresponding changes be made to corresponding transmitter(s).

Symbol-Based Power and Energy Considerations

The OTFS method also enables more sophisticated tradeoffs to be madebetween transmission distance, transmitter power, and information datarate than is possible to be made using conventional modulationtechniques. This increased flexibility arises in part because eachsymbol is generally spread over a larger number of intervals relative tothe case in which conventional techniques are employed. For example, inconventional time-division multiplexed communication systems the powerper symbol transmitted must be quite high because the symbol is beingtransmitted over only one time interval. In conventional spread spectrumcommunication systems, the symbol is being transmitted over essentiallyN intervals, and the power per interval is correspondingly less. Becausethe OTFS method transmits a bit or symbol of information over N²different modalities (e.g. waveforms, times), the power per modality ismuch less. Among other things, this means that the effect of impulsenoise, that would in general only impact a specific waveform over aspecific time interval, will be less. It also means that due toincreased number of signal transmission modalities (waveforms, times)enabled by the OTFS method, there are more degrees of freedom availableto optimize the signal to best correspond to the particularcommunications link impairment situation at hand.

Overview of OTFS Equalization

Attention is now directed to FIGS. 27-36, to which reference will bemade in describing various techniques for compensating for Doppler andfrequency shift within an OTFS communication system. Turning now to FIG.27, there is shown an exemplary process by which a receiver 2706compensates for various types of echo reflections or other channeldistortions through time deconvolution of a received signal in themanner described herein. In FIG. 27, wireless transmitter 2700 transmitsa complex cyclically time shifted and cyclically frequency shiftedwireless waveform 2702 in multiple directions using methods inaccordance with the above description. The wireless transmitter 2700could be realized using, for example, the OTFS transmitter 405 of FIG.4. Some of these signals 2704 go directly to the receiver 2706. Thereceiver 2706 can be, for example, the OTFS receiver 455 of FIG. 4.Other signals 2708 may be reflected by a wireless reflector, such as abuilding 2707. These “echo” reflections 2710 travel a longer distance toreach receiver 2706, and thus end up being time delayed. As a result,receiver 2706 receives a distorted signal 2712 that is the summation ofboth the original signal 2704 and the echo waveforms 2710.

Since a portion of the transmitted signal 2702 is a cyclically timeshifted waveform, a time deconvolution device 2714 at the receiver, suchas the post-equalizer 480 of FIG. 4, analyzes the cyclically timevarying patterns of the waveforms and effects appropriate compensation.In the embodiment of FIG. 27 this analysis may include a type of patternmatching or the equivalent and the decomposition of the distorted,received signal back into various time-shifted versions. Thesetime-shifted versions may include, for example, a first time-shiftedversion 2716 corresponding to direct signals 2704 and a secondtime-shifted version 2718 corresponding to the reflected signal 2710.The time deconvolution device 2714 may also determine the time-offset2720 necessary to cause the time delayed echo signal 2718, 2710 to matchup with the original or direct signal 2716, 2704. This time offset value2720, here called a time deconvolution parameter, may provide usefulinformation as to the relative position of the echo location(s) relativeto the transmitter 2700 and receiver 2706. This parameter may also helpthe system characterize some of the signal impairments that occurbetween the transmitter and receiver.

FIG. 28 shows an example of how transmitting both cyclically timeshifted waveforms and cyclically frequency shifted waveforms can beuseful to help a receiver 2806 (such as the OTFS receiver 455) effectboth time and frequency compensation of the received signal tocompensate for both echo reflections and frequency shifts—in thisexample Doppler effect frequency shifts. In FIG. 28, a moving wirelesstransmitter 2800, such as the OTFS transmitter 405, is againtransmitting a complex cyclically time shifted and cyclically frequencyshifted wireless waveform 2802 in multiple directions. To simplifypresentation, it is assumed that transmitter 2800 is movingperpendicular to receiver 2806 so that it is neither moving towards noraway from the receiver, and thus there are no Doppler frequency shiftsrelative to the receiver 2806. It is further assumed that thetransmitter 2800 is moving towards a wireless reflector, such as abuilding 2807, and thus the original wireless waveform 2802 will bemodified by Doppler effects, thereby shifting frequencies of thewaveform 2802 towards a higher frequency (blue shifted) relative to thereflector 2807.

Thus, the direct signals 2804 impinging upon the receiver 2806 will, inthis example, not be frequency shifted. However the Doppler-shiftedwireless signals 2808 that bounce off of the wireless reflector, hereagain building 2807, will echo off in a higher frequency shifted form.These higher frequency shifted “echo” reflections 2810 also still haveto travel a longer distance to reach receiver 2806, and thus also end upbeing time delayed as well. As a result, receiver 2806 receives a signal2812 that is distorted due to the summation of the direct signal 2804with the time and frequency shifted echo waveforms 2810.

However, as was described above, the OTFS techniques described hereinmay utilize the transmission of cyclically time shifted and frequencyshifted waveforms. Accordingly, a time and frequency deconvolutiondevice 2814 (alternatively a time and frequency adaptive equalizer suchas the OTFS demodulator 460 and the OTFS post-equalizer 480 of FIG. 4)within the receiver 2806 may evaluate the cyclically time varying andfrequency varying patterns of the waveforms in order to decompose suchwaveforms back into various time-shifted and frequency shifted versions.Included among such versions are a first version 2816 corresponding tothe direct signal 2804 and a second version 2818 corresponding to thefrequency shifted echo waveform 2810. In one embodiment this evaluationand decomposition may be effected using pattern matching or relatedtechniques. At the same time, the time and frequency deconvolutiondevice 2814 may also determine the above-referenced time deconvolutionparameter and a frequency offset value 2822, which may also be referredto herein as a frequency deconvolution parameter. These parameters mayprovide useful information as to the relative position of the echolocation(s) relative to the transmitter 2800 and the receiver 2806, andmay also enable characterization of certain of the signal impairmentsthat occur between the transmitter and receiver.

The net effect of both time and frequency deconvolutions, when appliedto transmitters, receivers, and echo sources that potentially exist atdifferent distances and velocities relative to each other, is to allowthe receiver to properly interpret the impaired signal. Here, even ifthe energy received in the primary signal is too low to permit properinterpretation, the energy from the time and/or frequency shiftedversions of the signals can be added to the primary signal upon theapplication of appropriate time and frequency offsets or deconvolutionparameters to signal versions, thereby resulting in a less noisy andmore reliable signal at the receiver. Additionally, the time andfrequency deconvolution parameters can contain useful information as tothe relative positions and velocities of the echo location(s) relativeto the transmitter and receiver, as well as the various velocitiesbetween the transmitter and receiver, and can also help the systemcharacterize some of the signal impairments that occur between thetransmitter and receiver.

Thus, in some embodiments the OTFS systems described herein may alsoprovide a method to provide an improved receiver where, due to eitherone or a combination of echo reflections and frequency offsets, multiplesignals associated with such reflections and offsets result in thereceiver receiving a time and/or frequency convolved composite signalrepresentative of time and/or frequency shifted versions of the N²summation-symbol-weighted cyclically time shifted and frequency shiftedwaveforms. Here, the improved receiver will further time and/orfrequency deconvolve the time and/or frequency convolved signal tocorrect for such echo reflections and the resulting time and/orfrequency offsets. This will result in both time and frequencydeconvolved results (i.e. signals, typically of much higher quality andlower signal to noise ratio), as well as various time and frequencydeconvolution parameters that, as will be discussed, are useful for anumber of other purposes.

Before going into a more detailed discussion of other applications,however, it is useful to first discuss the various waveforms in moredetail.

Embodiments of the OTFS systems and methods described herein generallyutilize waveforms produced by distributing a plurality of data symbolsinto one or more N×N symbol matrices, and using these one or more N×Nsymbol matrices to control the signal modulation of a transmitter. Here,for each N×N symbol matrix, the transmitter may use each data symbol toweight N waveforms, selected from an N²-sized set of all permutations ofN cyclically time shifted and N cyclically frequency shifted waveformsdetermined according to an encoding matrix U, thus producing Nsymbol-weighted cyclically time shifted and cyclically frequency shiftedwaveforms for each data symbol. This encoding matrix U is chosen to bean N×N unitary matrix that has a corresponding inverse decoding matrixU^(H). The method will further, for each data symbol in the N×N symbolmatrix, sum the N symbol-weighted cyclically time shifted and cyclicallyfrequency shifted waveforms, producing N² summation-symbol-weightedcyclically time shifted and cyclically frequency shifted waveforms. Thetransmitter will transmit these N² summation-symbol-weighted cyclicallytime shifted and cyclically frequency shifted waveforms, structured as Ncomposite waveforms, over any combination of N time blocks or frequencyblocks.

As discussed above, various waveforms can be used to transmit andreceive at least one frame of data [D] (composed of a matrix of up to N²data symbols or elements) over a communications link. Here each datasymbol may be assigned a unique waveform (designated a correspondingwaveform) derived from a basic waveform.

For example, the data symbols of the data matrix [D] may be spread overa range of cyclically varying time and frequency shifts by assigningeach data symbol to a unique waveform (corresponding waveform) which isderived from a basic waveform of length N time slices (in embodimentsdescribed herein the set of N time slices correspond to the timerequired to transmit this waveform, also referred to as a time block),with a data symbol specific combination of a time and frequency cyclicshift of this basic waveform.

In one embodiment each symbol in the frame of data [D] is multiplied byits corresponding waveform, producing a series of N² weighted uniquewaveforms. Over one spreading time interval (or time block interval),all N² weighted unique waveforms corresponding to each data symbol inthe fame of data [D] are simultaneously combined and transmitted.Further, a different unique basic waveform of length (or duration) ofone time block (N time slices) may be used for each consecutivetime-spreading interval (consecutive time block). Thus a differentunique basic waveform corresponding to one time block may be used foreach consecutive time-spreading interval, and this set of N uniquewaveforms generally forms an orthonormal basis. Essentially, each symbolof [D] is transmitted (in part) again and again either over all N timeblocks, or alternatively over some combination of time blocks andfrequency blocks (e.g. assigned frequency ranges).

To receive data over each block of time, the received signal iscorrelated with a corresponding set of all N² waveforms previouslyassigned to each data symbol by the transmitter for that specific timeblock. Upon performing this correlation, the receiver may produce aunique correlation score for each one of the 12 data symbols. Thisprocess will be repeated over some combination of time blocks andfrequency blocks until all N blocks are received. The original datamatrix [D] can thus be reconstructed by the receiver by summing, foreach data symbol, the correlation scores over N time blocks or frequencyblocks, and this summation of the correlation scores will reproduce the12 data symbols of the frame of data [D].

Note that in some embodiments, some of these N time blocks may betransmitted non-consecutively, or alternatively some of these N timeblocks may be frequency shifted to an entirely different frequencyrange, and transmitted in parallel with other time blocks from theoriginal set of N time blocks in order to speed up transmission time.This is discussed later and in more detail in reference to FIG. 29.

In order to enable focus to be directed to the underlying cyclicallytime shifted and cyclically shifted waveforms, detailed aspects of oneembodiment of the OTFS methods described above may be somewhatgeneralized and also discussed in simplified form. For example, theoperation of selecting from an N² set of all permutations of Ncyclically time shifted and N cyclically frequency shifted waveforms maycorrespond, at least in part, to an optional permutation operation P aswell as to the other steps discussed above. Additionally, the N² set ofall permutations of N cyclically time shifted and N cyclically frequencyshifted waveforms may be understood, for example, to be at leastpartially described by a Discrete Fourier transform (DFT) matrix or anInverse Discrete Fourier Transform matrix (IDFT). This DFT and IDFTmatrix can be used by the transmitter, for example, to take a sequenceof real or complex numbers and modulate them into a series of differentwaveforms.

Considering now a particular example, individual rows of a DFT matrix(e.g., the DFT matrix of FIG. 18) may each be used to generate a Fouriervector including set of N cyclically time-shifted and frequency-shiftedwaveforms. In general, the Fourier vectors may create complex sinusoidalwaveforms of the type:

$X_{j}^{k} = e^{(\frac{{{- i^{*}}2^{*}\pi^{*}j^{*}k})}{N}}$

where, for an N×N DFT matrix [X}, X_(j) ^(k) is the coefficient of theFourier vector in row k column j of the DFT matrix, and N is the numberof columns. The products of this Fourier vector may be considered torepresent one example of a manner in which the various time shifted andfrequency shifted waveforms suitable for use in the OTFS system may begenerated.

For example and as mentioned previously, FIG. 10 shows a diagram of oneexample of a cyclic convolution method that a transmitter can use toencode and transmit data. In FIG. 10, the sum of the various[b^(m)*X^(k)] components can also be termed a “composite waveform”. As aconsequence, in an embodiment consistent with FIG. 10 the full [D]matrix of symbols will ultimately be transmitted as N compositewaveforms.

Although previously discussed, FIG. 12 may also be understood to providea diagram of a cyclic deconvolution method capable of being used todecode received data. More specifically, particularly in the case where[U₁] is composed of a cyclically permuted Legendre number of length N,the process of deconvolving the data and reconstructing the data can beunderstood alternatively as being a cyclic deconvolution (cyclicdecoding) of the transmitted data previously convolved (encoded) by thetransmitter as described in reference to FIG. 10. In the embodiment ofFIG. 12, the ˜d⁰, ˜d^(k), ˜d^(N−1) elements represent the reconstructedsymbols (symbols) of the data vector 1200 component of the [D] matrix(corresponding to the transmitted data vector 1000), the b^(m)coefficients again represent the base vector 1002 components of the [U₁]matrix, and the X_(j) ^(k) coefficients can again be understood asrepresenting the Fourier vector 1004 components of the [U₂] matrix. Here(R^(m)) 1202 is a portion of the accumulated signal 1010 received anddemodulated by the receiver.

As described above with reference to FIGS. 24-26, different tilingschemes for proportioning the rows (frequency offsets) and columns (timeoffsets) of the data matrix [D] can be utilized to provide for multipleusers to transmit data over multiple time/frequency offset blocks in thesame data matrix [D]. These tiling schemes can be utilized differentlydepending on the type(s) of motion and reflected signals and theresulting time and frequency offsets that a transmitter and receiver areexperiencing. Some exemplary methods for utilizing differenttime/frequency blocks will now be described with reference to FIGS.29-30.

Referring now to FIG. 29, there are shown various transmitted waveformblocks 2900 can be transmitted as a series of N consecutive time blocks(i.e. no other blocks in between). These consecutive time blocks caneither be a contiguous series 2902 (i.e. with minimal or no time gaps inbetween various waveform blocks) or they can be a sparsely contiguousseries 2904 (i.e. with time gaps between the various waveform bocks,which may in some embodiments be used for synchronization, hand shaking,listening for other user's transmitters, channel assessment and otherpurposes. Alternatively, the various waveform time blocks can betransmitted either time-interleaved with the blocks from one or moredifferent symbol matrices 2906, 2908 (which in some cases may be from adifferent transmitter) in a contiguous or sparse interleaved manner asshown in series 2910.

As yet another alternative, some of the various waveform time blocks maybe frequency transposed to entirely different frequency bands or ranges2912, 2914, 2916. This can speed up transmission time, because nowmultiple waveform time blocks can now be transmitted at the same time asdifferent frequency blocks. As shown in time/frequency offset tiles 2918and 2920, such multiple frequency band transmissions can also be done ona contiguous, sparse contiguous, contiguous interleaved, or sparsecontiguous interleaved manner. Here 2922 and 2928 represent one timeblock at a first frequency range 2912, and 2924 and 2930 represent thenext time block at the frequency range 2912. Here the various frequencyranges 2912, 2914, and 2916 can be formed, as will be described shortly,by modulating the signal according to different frequency carrier waves.Thus, for example, frequency range or band 2912 might be transmitted bymodulating a 1 GHz frequency carrier wave, frequency range or band 2914might be transmitted by modulating a 1.3 GHz frequency carrier wave, andband 2915 might be transmitted by modulating a 1.6 GHz frequency carrierwave, and so on.

Stated differently, the N composite waveforms, themselves derived fromthe previously discussed N² summation-symbol-weighted cyclically timeshifted and cyclically frequency shifted waveforms, may be transmittedover at least N time blocks. These N time blocks may be eithertransmitted consecutively in time (e.g. 2902, 2904) or alternativelytransmitted time-interleaved with the N time blocks from a second anddifferent N×N symbol matrix.

FIG. 30 shows that the various composite waveform blocks transmitted bythe transmitter can be either transmitted as shorter duration timeblocks over one or more wider frequency ranges, or as longer durationtime blocks over one or more narrower frequency ranges. That is, FIG. 30shows exemplary tradeoffs between frequency bandwidth and time madeavailable through use of embodiments of the OTFS method. Whereas intime/frequency tile 2940, the available bandwidth for each frequencyrange 2912, 2914, and 2916 is relatively large, in 2942, the availablebandwidth for each frequency range 2932, 2934 and 2936 is considerablyless. Here, the OTFS scheme can compensate for narrower frequency rangesby allowing more time per time block. Thus where as in time/frequencytile 2940, with high bandwidth available, the time blocks 2922 and 2924can be shorter, in time/frequency tile 2942, with lower bandwidthavailable, the time blocks 2926 for transmitting the composite waveformis longer.

For both FIGS. 29 and 30 then, if there is only one fundamental carrierfrequency, then all N blocks are transmitted consecutively in time as Ntime blocks. If there are less than N multiple fundamental carrierfrequencies available, then all N blocks can be transmitted as somecombination of N time blocks and N frequency blocks. If there are N ormore fundamental frequencies available, then all N blocks can betransmitted over the duration of 1 time block as N frequency blocks.

Attention is now again directed to FIG. 21, to which reference will bemade in describing an exemplary pre-equalization scheme. As wasdescribed previously, the transmitter 2100 is configured to transmit aseries of N consecutive waveform time blocks where each time blockencompasses a set of N time slices. During every successive time slice,one element from the OTFS matrix 2108 can be used to control themodulation circuit 2104. As was also previously discussed, themodulation scheme may be one in which the element will be separated intoits real and imaginary components, chopped and filtered, and then usedto control the operation of a sin and cosine generator, producing acomposite analog waveform 2120. The net, effect, by the time that theentire original N×N data symbol matrix [D] is transmitted, is totransmit the data in the form of N² summation-symbol-weighted cyclicallytime shifted and cyclically frequency shifted waveforms, structured as Ncomposite waveforms.

In some embodiments the transmitter 2100 may further implement apre-equalization operation, typically performed by the pre-equalizer 410of FIG. 4, which involves processing the [D] matrix prior to providingit to the analog modulation circuit 2102. When this pre-equalizationoperation is performed, the transmitter 2100 outputs pre-equalized OTFSsignals 2130; otherwise, the transmitter simply outputs the OTFS signals2120. The pre-equalization operation may be performed when, for example,the receiver in communication with the transmitter 2100 detects that anOTFS signal 2120 has been subjected to specific echo reflections and/orfrequency shifts. Upon so detecting such echo reflections and/orfrequency shifts, the receiver may transmit corrective information tothe transmitter pertinent to such reflections and shifts. Thepre-equalizer 410 may then shape subsequently-transmitted pre-equalizedOTFS signals so as to compensate for these echo reflections and/orfrequency shift. Thus, for example, if the receiver detects an echodelay, the pre-equalizer 410 may send the signal with an anti-echocancellation waveform. Similarly, if the receiver detects a frequencyshift, the pre-equalizer 410 can introduce a compensatory reversefrequency shift into the transmitted pre-equalization signal 2130.

FIG. 31 illustrates exemplary receiver processing section 3110 operativeto compensate for the effects of echo reflections and frequency shifts.Referring to FIG. 31, the receiver processing section 3110 includes acyclic deconvolution processing block 3106 and an equalizer 3102. Theequalizer 3102 performs a series of math operations and outputsequalization parameters 3108 that can also give information pertainingto the extent to which the echo reflections and frequency shiftsdistorted the underlying signal. The equalizer 3102A can be, forexample, an adaptive equalizer.

In FIG. 31, it is assumed that the composite transmitted waveform has,since transmission, been distorted by various echo reflections and/orfrequency shifts as previously shown in FIGS. 27 and 28. This produces adistorted waveform 3100, which for simplicity is represented through asimple echo reflection delayed distortion. In FIG. 31, equalizer 3102 isconfigured to reduce or substantially eliminate such distortion byanalyzing the distorted waveform 3100 and, assisted by the knowledgethat the original composite waveform was made up of N cyclically timeshifted and N cyclically frequency shifted waveforms, determine whatsort of time offsets and frequency offsets will best deconvolvedistorted waveform 3100 back into a close representation of the originalwaveform, which is represented in FIG. 31 as deconvolved waveform 3104.The equalization operations performed by equalizer 3102 may alternatelybe carried out by the cyclic deconvolution device 3106.

In one embodiment the equalizer 3102 produces a set of equalizationparameters 3108 during the process of equalizing the distorted waveform.For example, in the simple case where the original waveform wasdistorted by only a single echo reflection offset by time t_(offset),and by the time the original waveform and the t_(offset) echo waveformreach the receiver, the resulting distorted signal 3100 may be, forexample, about 90% original waveform and 10% t_(offset) echo waveform,then the equalization parameters 3108 can output both the 90% originaland 10% echo signal mix, as well as the t_(offset) value. Typically, ofcourse, the actual distorted signal 3100 could consist of a number ofvarious time and frequency offset components, and here again, inaddition to cleaning this distortion, the equalizer 3102 can also reportthe various time offsets, frequency offsets, and percentage mix of thevarious components of signal 3100 to the transmitter and/or thereceiver.

As previously discussed in FIGS. 29 and 30, the various compositewaveforms in the N time blocks can be transmitted in various ways. Inaddition to time consecutive transmission, i.e. a first block, followed(often by a time gap which may optionally be used for handshaking orother control signals) by a second time block and then a third timeblock, the various blocks of composite waveforms can be transmitted byother schemes.

In some embodiments, for example in network systems where there may bemultiple transmitters and potentially also multiple receivers, it may beuseful to transmit the data from the various transmitters using morethan one encoding method. Here, for example, a first set of N timeblocks may transmit data symbols originating from a first N×N symbolmatrix from a first transmitter using a first unitary matrix [U₁]. Asecond set of N time blocks may transmit data symbols originating from asecond N×N symbol matrix from a second transmitter using a secondunitary matrix [U₂]. Depending on the embodiment, [U₁] and [U₂] may beidentical or different. Because the signals originating from the firsttransmitter may encounter different impairments (e.g. different echoreflections, different frequency shifts), some schemes of cyclicallytime shifted and cyclically frequency shifted waveforms may operatebetter than others. Thus, these waveforms, as well as the unitarymatrices [U₁] and [U₂], may be selected based on the characteristics ofthese particular echo reflections, frequency offsets, and other signalimpairments of the system and environment of the first transmitter, thesecond transmitter and/or the receiver.

As an example, a receiver configured to implement equalization inaccordance with FIG. 31 may, based upon the equalization parameters 3108which it derives, elect to propose an alternative set of cyclically timeshifted and cyclically frequency shifted waveforms intended to providesuperior operation in view of the current environment and conditionsexperienced by such receiver. In this case the receiver could transmitthis proposal (or command) to the corresponding transmitter(s). Thistype of “handshaking” can be done using any type of signal transmissionand encoding scheme desired. Thus in a multiple transmitter and receiverenvironment, each transmitter may attempt to optimize its signal so thatits intended receiver is best able to cope with the impairments uniqueto communication between the transmitter and receiver over thecommunications channel therebetween.

In some cases, before transmitting large amounts of data, or any time asdesired, a given transmitter and receiver may choose to more directlytest the various echo reflections, frequency shifts, and otherimpairments of the transmitter and receiver's system and environment.This can be done, by, for example having the transmitter send a testsignal where the plurality of data symbols are selected to be testsymbols known to the receiver (e.g., the receiver may have stored arecord of these particular test symbols). Since in this case thereceiver will be aware of exactly what sort of signal it should receivein the absence of any impairment, the equalizer 3102 will generally beable to provide even more accurate time and frequency equalizationparameters 3108 for use by the receiver relative to the case in whichthe receiver lacks such awareness. Thus, in this case the equalizationparameters provide even more accurate information relating to thecharacteristics of the echo reflections, frequency offsets, and othersignal impairments of the system and environment of the applicabletransmitter(s) and the receiver. This more accurate information may beused by the receiver to suggest or command that the applicabletransmitter(s) shift to use of communications schemes (e.g., to Umatrices) more suitable to the present situation.

In some embodiments, when the transmitter is a wireless transmitter andthe receiver is a wireless receiver, and the frequency offsets arecaused by Doppler effects, the more accurate determination of thedeconvolution parameters, i.e. the characteristics of the echoreflections and frequency offsets, can be used to determine the locationand velocity of at least one object in the environment of thetransmitter and receiver.

Examples of OTFS Equalization Techniques

This section includes a description of a number of exemplary OTFSequalization techniques capable of being implemented consistent with thegeneral OTFS equalization approach and apparatus discussed above.However, prior to describing such exemplary techniques, a summary ofaspects of transmission and reception of OTFS-modulated signals is givenin order to provide an appropriate context for discussion of these OTFSequalization techniques.

Turning now to such a summary of OTFS signal transmission and reception,consider the case in which a microprocessor-controlled transmitterpackages a series of different symbols “d” (e.g. d₁, d₂, d₃ . . . ) fortransmission by repackaging or distributing the symbols into variouselements of various N×N matrices [D]. In one implementation suchdistribution may, for example, include assigning d₁ to the first row andfirst column of the [D] matrix (e.g. d₁=d_(0,0)), d₂ to the first row,second column of the [D] matrix (e.g. d₂=d_(0,1)) and so on until allN×N symbols of the [D] matrix are full. Here, if the transmitter runsout of “d” symbols to transmit, the remaining [D] matrix elements can beset to be 0 or other value indicative of a null entry.

The various primary waveforms used as the primary basis for transmittingdata, which here will be called “tones” to show that these waveformshave a characteristic sinusoid shape, can be described by an N×N InverseDiscrete Fourier Transform (IDFT) matrix [W], where for each element win [W],

$w_{j,k} = e^{\frac{{i\; 2\pi\;{jk}}\;}{N}}$or alternatively w_(j,k)=e^(ijθ) ^(k) or w_(j,k)=[e^(iθ) ^(k) ]^(j).Thus the individual data elements d in [D] are transformed anddistributed as a combination of various fundamental tones w by a matrixmultiplication operation [W]*[D], producing a tone transformed anddistributed form of the data matrix, here described by the N×N matrix[A], where [A]=[W]*[D].

To produce N cyclically time shifted and N cyclically frequency shiftedwaveforms, the tone transformed and distributed data matrix [A] is thenitself further permuted by modular arithmetic or “clock” arithmetic,thereby creating an N×N matrix [B], including each element b of [B],b_(i,j)=a_(i,(i+j)mod N). This can alternatively be expressed as[B]=Permute([A])=P(IDFT*[D]). Thus the clock arithmetic controls thepattern of cyclic time and frequency shifts.

The previously described unitary matrix [U] can then be used to operateon [B], producing an N×N transmit matrix [T], where [T]=[U]*[B], thusproducing an N² sized set of all permutations of N cyclically timeshifted and N cyclically frequency shifted waveforms determinedaccording to an encoding matrix [U].

Put alternatively, the N×N transmit matrix [T]=[U]*P(IDFT*[D]).

Then, typically on a per column basis, each individual column of N isused to further modulate a frequency carrier wave (e.g. if transmittingin a range of frequencies around 1 GHz, the carrier wave will be set at1 GHz). In this case each N-element column of the N×N matrix [T]produces N symbol-weighted cyclically time shifted and cyclicallyfrequency shifted waveforms for each data symbol. Effectively then, thetransmitter is transmitting the sum of the N symbol-weighted cyclicallytime shifted and cyclically frequency shifted waveforms from one columnof [T] at a time as, for example, a composite waveform over a time blockof data. Alternatively, the transmitter could instead use a differentfrequency carrier wave for the different columns of [T], and thus forexample transmit one column of [T] over one frequency carrier wave, andsimultaneously transmit a different column of [T] over a differentfrequency carrier wave, thus transmitting more data at the same time,although of course using more bandwidth to do so. This alternativemethod of using different frequency carrier waves to transmit more thanone column of [T] at the same time will be referred to as frequencyblocks, where each frequency carrier wave is considered its ownfrequency block.

Thus, since the N×N matrix [T] has N columns, the transmitter willtransmit the N² summation-symbol-weighted cyclically time shifted andcyclically frequency shifted waveforms, structured as N compositewaveforms, over any combination of N time blocks or frequency blocks, aspreviously shown in FIGS. 29 and 30.

On the receiver side, the transmit process is essentially reversed.Here, for example, a microprocessor controlled receiver would of coursereceive the various columns [T] (e.g., receive the N compositewaveforms, also known as the N symbol-weighted cyclically time shiftedand cyclically frequency shifted waveforms) over various time blocks orfrequency blocks as desired for that particular application. In cases inwhich sufficient bandwidth is available and time is of the essence, thetransmitter may transmit the data as multiple frequency blocks overmultiple frequency carrier waves. On the other hand, if availablebandwidth is more limited, and/or time (latency) is less critical, thenthe transmit will transmit and the receiver will receive over multipletime blocks instead.

During operation the receiver may effectively tune into the one or morefrequency carrier waves, and over the number of time and frequencyblocks set for the particular application, eventually receive the dataor coefficients from the original N×N transmitted matrix [T] as an N×Nreceive matrix [R]. In the general case [R] will be similar to [T], butmay not be identical due to the existence of various impairments betweenthe transmitter and receiver.

The microprocessor controlled receiver then reverses the transmitprocess as a series of steps that mimic, in reverse, the originaltransmission process. The N×N receive matrix [R] is first decoded byinverse decoding matrix [U^(H)], producing an approximate version of theoriginal permutation matrix [B], here called [B^(R)], where[B^(R)]=([U^(H)]*[R]).

The receiver then does an inverse clock operation to back out the datafrom the cyclically time shifted and cyclically frequency shiftedwaveforms (or tones) by doing an inverse modular mathematics or inverseclock arithmetic operation on the elements of the N×N [B^(R)] matrix,producing, for each element b^(R) of the N×N [B^(R)] matrix, a_(i,j)^(R)=b_(i,(j−i)mod N) ^(R). This produces a de-cyclically time shiftedand de-cyclically frequency shifted version of the tone transformed anddistributed form of the data matrix [A], which may hereinafter bereferred to as [A^(R)]. Put alternatively, [A^(R)]=Inverse Permute([B^(R)]), or [A^(R)]=P⁻¹([U^(H))]*[R]).

The receiver then further extracts at least an approximation of theoriginal data symbols d from the [A^(R)] matrix by analyzing the [A]matrix using an N×N Discrete Fourier Transform matrix DFT of theoriginal Inverse Fourier Transform matrix (IDFT).

Here, for each received symbol d^(R), the d^(R) are elements of the N×Nreceived data matrix [D^(R)] where [D^(R)]=DFT*A^(R), or alternatively[D^(R)]=DFT*P⁻¹([U^(H)]*[R]).

Thus the original N² summation-symbol-weighted cyclically time shiftedand cyclically frequency shifted waveforms are subsequently received bya receiver which is controlled by the corresponding decoding matrixU^(H) (also represented as [U^(H)]) The processor of the receiver usesthis decoding matrix [U^(H)] to reconstruct the various transmittedsymbols “d” in the one or more originally transmitted N×N symbolmatrices [D] (or at least an approximation of these transmittedsymbols).

Turning now to a discussion of various exemplary OTFS equalizationtechniques, there exist at least several general approaches capable ofbeing used correct for distortions caused by the signal impairmenteffects of echo reflections and frequency shifts. One approach leveragesthe fact that the cyclically time shifted and cyclically frequencyshifted waveforms or “tones” form a predictable time-frequency pattern.In this scheme a deconvolution device situated at the receiver's frontend may be straightforwardly configured to recognize these patterns, aswell as the echo-reflected and frequency shifted versions of thesepatterns, and perform the appropriate deconvolutions by a patternrecognition process. Alternatively the distortions may be mathematicallycorrected using software routines, executed by the receiver's processor,designed to essentially determine the various echo reflected andfrequency shifting effects, and solve for these effects. As a thirdalternative, once, by either process, the receiver determines the timeand frequency equalization parameters of the communication media'sparticular time and frequency distortions, the receiver may transmit acommand to the transmitter to instruct the transmitter to essentiallypre-compensate or pre-encode, e.g., by using a pre-equalizer such as thepre-equalizer 410 of FIG. 4, for these effects. That is, if for examplethe receiver detects an echo, the transmitter can be instructed totransmit in a manner that offsets this echo, and so on.

FIG. 32A illustrates an exemplary system in which echo reflections andfrequency shifts (e.g., Doppler shifts caused by motion) of a channel Hecan blur or be distorted by additive noise 3202. The time and frequencydistortions can be modeled as a 2-dimensional filter He acting on thedata array. This filter He represents, for example, the presence ofmultiple echoes with time delays and Doppler shifts. To reduce thesedistortions, the signal can be pre-equalized, e.g., using thepre-equalizer 3208, before the signal 3200 is transmitted by thetransmitter 3204 over the channel to the receiver 3205 and subsequentlypost-equalized, using the post-equalizer 3206, after the D^(R) matrixhas been recovered at 3206. This equalization process may be performedby, for example, using digital processing techniques. The equalized formof the received D matrix, which ideally will completely reproduce theoriginal D matrix, may be referred to hereinafter as D_(eq).

FIG. 32B shows an example of an adaptive linear equalizer 3240 that maybe used to implement the post-equalizer 3206 in order to correct forsuch distortions. The adaptive linear equalizer 3240, which may also beused as the equalizer 3102, may operate according to the function:

${Y(k)} = {{\sum\limits_{L = {Lc}}^{Rc}\;{{C(l)}^{*}{X\left( {k - l} \right)}}} + {\eta(k)}}$

Mathematical Underpinnings of Two Dimensional Equalization

An exemplary equalization mechanism associated with the OTFS modulation,which is inherently two dimensional, is discussed below. This is incontrast to its one dimensional counterpart in conventional modulationschemes such as, for example, OFDM and TDMA.

Assume that the input symbol stream provided to an OTFS transmitter is adigital function X∈

(R_(d)×R_(d)) with values in specific finite constellation

⊂

(either QPSK or higher QAM's, for example). This transmitter modulatesthis input stream into an analog signal Φ^(Tx,Pass), which is thentransmitted. During the transmission Φ^(Tx,Pass) undergo a multipathchannel distortion. The distorted passband signal Φ^(Tx,Pass) arrives atthe OTFS receiver and is demodulated back into a digital function Y∈

(R_(d)×R_(d)), which may be referred to herein as the output stream. Thelocality properties of OTFS modulation imply that the net effect of themultipath channel distortion is given by a cyclic two dimensionalconvolution with a two dimensional channel impulse response. See FIGS.53 and 54.

Referring to FIG. 53, an illustration is provided of a two-dimensionalchannel impulse. Smear along the time axis represents multipathreflections causing time delay while smear along the frequency axisrepresents multipath reflectors causing Doppler shifts. In FIGS.54A-54C, input and output streams are depicted after two-dimensionalchannel distortion. Specifically, FIG. 54A represents thetwo-dimensional channel impulse, FIG. 54B represents a portion of theinput stream and FIG. 54B depicts the same portion after convolutionwith the channel and additive noise.

In what follows an appropriate equalization mechanism will be described.To this end, it will be convenient to enumerate the elements of thedigital time axis by 0, 1, . . . , N−1 and to consider the input andoutput streams X and Y, respectfully, as sequences of functions:X=(X(k)∈

(R _(d)):k=0, . . . ,N−1),Y=(k)∈

(R _(d)):k=0,. . . ,N−1),where X(k)(i)=X(k, i) and Y(k)(i)=Y(k, i), for every k=0, . . . , N−1and i∈R_(d).

Furthermore, for purposes of explanation it will be assumed that thetime index k is infinite in both directions, that is, k∈

, the digital time direction is linear, and the digital frequencydirection is cyclic. Under these conventions the relation between theoutput stream and the input stream can be expressed by the followingEquation (1.1):

$\begin{matrix}{{{Y(k)} = {{\sum\limits_{l = L_{C}}^{R_{C}}\;{{C(l)}*{X\left( {k - l} \right)}}} + {(k)}}},} & 1.1\end{matrix}$

where:

-   -   C=(C(l)∈        (R_(d)): l=L_(C), . . . , R_(C)) are the channel impulse taps.        Typically, L_(C)∈        ^(<0) and R_(C)∈        ^(>0). The number n_(C)=R_(C)−L_(C)+1 may be referred to herein        as the memory length of the channel. The operation * in (4.1)        stands for one dimensional cyclic convolution on the ring R_(d).    -   (k)∈(0, N₀−Id_(N)) is a complex Gaussian N dimensional vector        with mean 0 and covariance matrix N₀·Id, representing the white        Gaussian noise.

Referring now to FIG. 32C, there is shown an exemplary adaptive decisionfeedback equalizer 3250 capable of being utilized as the equalizer 3102(FIG. 31). The adaptive decision feedback equalizer 3250 both shifts theecho and frequency shifted signals on top of the main signal in aforward feedback process 3210, and also then uses feedback signalcancelation methods to further remove any residual echo and frequencyshifted signals in 3312. The method then effectively rounds theresulting signals to discrete values.

The adaptive decision feedback equalizer 3250 may, in some embodiments,operate according to the function:

${X^{s}(k)} = {{\sum\limits_{l = L_{F}}^{R_{F}}\;{{F(l)}^{*}{Y\left( {k + l} \right)}}} - {\sum\limits_{l - L_{B}}^{- 1}\;{{B(l)}^{*}{X^{h}\left( {k + l} \right)}}}}$Where   X^(H)(k) = Q(X^(s)(k))

Decision Feedback Least Mean Square Estimator (DF-LMS) With LockedCarrier Frequency

An exemplary decision feedback LMS equalizer adapted to the relationexpressed in Equation (1.1) will now be described under the conditionthat the carrier frequency is locked between the transmitter andreceiver, that is, W_(Tx)=W_(Rx). An adaptation of the equalizer underthe condition of the existence of a non-zero discrepancy, i.e., ΔW≠0,will subsequently be described. In one aspect, the equalizerincorporates a forward filter and a feedback filter as follows:F=(F(l)∈

(R _(d)): l=L _(F) , . . . ,R _(F)),  Forward filter:B=(B(l)∈(R _(d)): l=L _(B), . . . ,−1),  Feedback filter:where, typically L_(F), L_(B)∈

^(<0) and R_(F)∈

⁰ satisfy L_(F), L_(B)≥L_(C) and R_(F)

R_(C). In fact, both filters depend on additional parameter k∈

designating the present point on the digital time axis, hence, thecomplete notation for the filter taps is F_(k) (l) and B_(k) (l).However, for the sake of presentation this additional index willgenerally be omitted and will only be included when necessary. A softestimator is defined as follows:

$\begin{matrix}{{X^{s}(k)} = {{\sum\limits_{l = L_{F}}^{R_{F}}\;{{F(l)}*{Y\left( {k + l} \right)}}} - {\sum\limits_{l - L_{B}}^{- 1}\;{{B(l)}*{{X^{h}\left( {k + l} \right)}.}}}}} & 1.2\end{matrix}$where X^(h)(k+l) is the past hard estimation of the past data vector X(k+l), l=L_(B), . . . , −1, defined as the quantization X^(h) (k)=Q(X⁸(k)), that is:

$\begin{matrix}{{{X^{h}(k)}(i)} = {\arg\;\underset{p \in C}{\mspace{11mu}\min\;}{{{{{X^{s}(k)}(i)} - p}}.}}} & 1.3\end{matrix}$

Computation of Initial Forward and Feedback Filter Taps

In one aspect, a closed formula may be used to determine the forward andfeedback filter taps of the decision feedback equalizer expressed interms of the channel impulse response. In this case the forward filtertaps are computed without regard to feedback and then the feedbackfilter taps are determined.

Computation of the Forward Filter Taps

First fix k=0 and let X⁸ denote the following soft estimator for thevector X (0), which depends only upon the forward filter taps:

$\begin{matrix}{X^{s} = {\sum\limits_{l = L_{F}}^{R_{F}}\;{{F(l)}*{{Y(l)}.}}}} & 1.7\end{matrix}$

In what follows it is assumed that X(k)˜

(0, P·Id_(N)), for every k∈

. Later this condition may be replaced by the condition that X(k)˜

(0, P·Id_(N)), for k≥0 and X(k)=0 for k<0, which is more adapted to thechoice of X⁸. We denote by Err=Err(0) the soft error term:Err=X ^(s) −X(0).  1.8

We consider the cost function:U(F)=

∥Err∥² =

∥X ^(s) −X(0)∥²,where the expectation is taken over the probability distribution of theinput stream X and the additive white Gaussian noise

. The optimal filter F^(opt) is defined as:

${F^{opt} = {\arg\mspace{11mu}{\min\limits_{F}\;{U(F)}}}},$therefore it satisfies the following system of linear equations:∇_(F(l)) U(F ^(opt))=0,l=L _(F) , . . . ,R _(F).  1.9

The formula for the gradient ∇_(F(l))U is an averaged version of (1.6),that is:

$\begin{matrix}\begin{matrix}{{\nabla_{F{(l)}}U} = {\;\left\lbrack {{Err}*{Y(l)}^{\bigstar}} \right\rbrack}} \\{= {\;\left\lbrack {\left( {X^{s} - {X(0)}} \right)*{Y(l)}^{\bigstar}} \right\rbrack}} \\{= {{\;\left\lbrack {X^{s}*{Y(l)}^{\bigstar}} \right\rbrack} - {{\;\left\lbrack {{X(0)}*{Y(l)}^{\bigstar}} \right\rbrack}.}}}\end{matrix} & 1.10\end{matrix}$

We first compute the term

[X(0)*Y(l)★] and then the term

[X^(s)*Y(l)★]. Developing the expression X(0)*Y(l))★ we obtain:

${{X(0)}*{Y(l)}^{\bigstar}} = {\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{{X(0)}*{X\left( {l - l^{\prime}} \right)}^{\bigstar}*{{C\left( l^{\prime} \right)}^{\bigstar}.}}}$

We observe that

[X(0)*X(k)★]=0 when l≠0 and

[X(0)*X(0)*X(0)★]=NP·δ_(w=0), hence we conclude that:

[X(0)*Y(l)^(★) ]=NP·δ _(w=0) *C(l)^(★) =NP·C(l)^(★).  1.11

Next, we compute the term

[X^(s)*Y(l)★]:

$\begin{matrix}{{\left\lbrack {X^{s}*{Y(l)}^{\bigstar}} \right\rbrack} = {\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{{F\left( l^{\prime} \right)}*{{\left\lbrack {{Y\left( l^{\prime} \right)}*{Y(l)}} \right\rbrack}.}}}} & 1.12\end{matrix}$

Developing the expression for Y(l′)*Y(l) we obtain:

$\begin{matrix}\begin{matrix}{{{Y\left( l^{\prime} \right)}*{Y(l)}} = {\left( {{\sum\limits_{r = L_{C}}^{R_{C}}\;{{C(r)}*{X\left( {l^{\prime} - r} \right)}}} + {\left( l^{\prime} \right)}} \right)*}} \\{\left( {{\sum\limits_{s = L_{C}}^{R_{C}}\;{{C(s)}*{X\left( {l - s} \right)}}} + {(l)}} \right)} \\{= {{\sum\limits_{s = L_{C}}^{R_{C}}\;{\sum\limits_{r = L_{C}}^{R_{C}}\;{{C(s)}^{\bigstar}*{C(r)}*{X\left( {l - s} \right)}*{X\left( {l^{\prime} - r} \right)}}}} +}} \\{{\left( l^{\prime} \right)*(l)} + {{Additional}\mspace{14mu}{{Terms}.}}}\end{matrix} & 1.13\end{matrix}$

Denote

${R\left( {l,l^{\prime}} \right)} = {\frac{1}{NP} \cdot {{\left\lbrack {{Y\left( l^{\prime} \right)}*{Y(l)}} \right\rbrack}.}}$Taking the expectation of both sides of (1.13) we obtain the followingexplicit formula for R(l, l′):

$\begin{matrix}{{R\left( {l,l^{\prime}} \right)} = \left\{ {\begin{matrix}{\sum\limits_{\underset{{s - r} = {l - l^{\prime}}}{s,{r = L_{C}}}}^{R_{C}}\;{{C(r)}*{C(s)}^{\bigstar}}} & {l \neq l^{\prime}} \\{{\sum\limits_{s = L_{C}^{\prime}}^{R_{C}}\;{{C(s)}*{C(s)}^{\bigstar}}} + {\frac{1}{SNR} \cdot \delta_{w = 0}}} & {l = l^{\prime}}\end{matrix},} \right.} & 1.14\end{matrix}$where in the computation of R(l,l′) we use the following conditions onthe mean of the specific terms in (1.13):

[X(k)*X(k′)]=NP·δ _(k=k′),

[

(k)*

(k′)]=NN ₀·δ_(k=k′),

[Additional Terms]=0.

Combining (1.9), (1.10), (1.11) and (1.12) we conclude that the optimalfilter F^(opt) satisfies the following system of linear equations:

$\begin{matrix}{{{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{{R\left( {l,l^{\prime}} \right)}*{F^{opt}\left( l^{\prime} \right)}}} = {C(l)}^{\bigstar}},{l = L_{F}},\ldots\mspace{11mu},{R_{F}.}} & 1.15\end{matrix}$

Finally, system (1.15) can be reduced to N systems ofn_(F)=R_(F)−L_(F)=1 scalar equations as follows. Applying a DFT to bothsides of (1.15) we obtain:

$\begin{matrix}{{{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\; \cdot} = =},{l = L_{F}},\ldots\mspace{11mu},{R_{F}.}} & 1.16\end{matrix}$where (

) stands for the DFT of the corresponding function and · stands forpointwise multiplication of functions in

(R_(d)) as we recall that DFT interchanges convolution with pointwisemultiplication and ★ with complex conjugation. Now observe that eachfunction valued equation in (1.16) decouples into n_(F) scalar valuedequations by evaluating both sides on each element of the ring R_(d).Explicitly, if we number the elements in R_(d) by 0, 1, 2, . . . , N−1we end up with the following scalar valued system of equations:

∑ l ′ = L F R F ⁢ ⁢ ⁢ ( i ) · ⁢ ( i ) = _ ⁢ ( i ) , l = L F , … ⁢ , R F , ⁢1.17for every i=0, . . . , N−1. In more concrete matrix form (1.17) lookslike:

( R ⁢ F ) ⁢ ( i ) · · R ⁢ F ) ⁢ ( i ) · · · · · · · · R ⁡ ( F ) ⁢ ( i ) · · R ⁡( F ) ⁢ ( i ) ) ⁢ ( ⁢ ( i ) · · ) ⁢ ( i ) ) = ( ) ⁢ ( i ) · · ⁢ ( i ) ) , 1.18for every i=0, . . . , N−1. We conclude the discussion by consideringthe case when the input stream satisfies X (k)=0 for k<0 which is thecase adapted to (feedback) subtraction of the past interference. In thisscenario, the optimal forward filter F^(opt) satisfies a system of theform (1.15) with the “matrix coefficients” R(l,l′) taking the form:

$\begin{matrix}{{R\left( {l,l^{\prime}} \right)} = \left\{ {\begin{matrix}{\underset{{s - r} = {l - l^{\prime}}}{\sum\limits_{s = L_{C}}^{\min{\{{l,R_{C}}\}}}\;\sum\limits_{r = L_{C}}^{\min{\{{l,R_{C}}\}}}}\;{C(s)}^{\bigstar}*{C(r)}} & {l \neq l^{\prime}} \\{{\sum\limits_{s = L_{C}}^{\min{\{{l,R_{C}}\}}}\;{{C(s)}^{\bigstar}*{C(s)}}} + {\frac{1}{SNR} \cdot \delta_{w = 0}}} & {l = l^{\prime}}\end{matrix},} \right.} & 1.19\end{matrix}$

Computation of the Feedback Filter Taps

The optimal feedback filter taps B^(opt) (l), l=L_(B), . . . , −1 can becomputed from the forward and channel taps according to the followingformula:

$\begin{matrix}{{B^{opt}(l)} = {\sum\limits_{l^{\prime} = L_{C}}^{R_{C}}\;{{F^{opt}\left( {l^{\prime} + l} \right)}*{{C\left( l^{\prime} \right)}.}}}} & 1.20\end{matrix}$

The justification of Formula (1.20) proceeds as follows. Fix an inputvector X(l₀) for some specific l₀=L_(B), . . . −1. Subtracting itsinterference C(l′)*X(l₀) from each term Y (l₀+l′), we obtain an“interference free” sequence {tilde over (Y)}(l), l=L_(F) . . . , R_(F).Now, applying the forward filter F^(opt) to the sequence {tilde over(Y)}(l) we obtain an estimator for X(0) given by

${X^{s} = {{\sum\limits_{l = L_{F}}^{R}\;{{F^{opt}(l)}*{X(l)}}} - {{B^{opt}\left( l_{0} \right)}*{X\left( l_{0} \right)}}}},$which concludes the justification.

Computation of Optimal Initial Forward and Feedback Filter Taps

In an alternative aspect, a closed formula of the optimal forward andfeedback filter taps of the decision feedback equalizer may be expressedin terms of the channel impulse response. In this regard we conduct thecomputation in the stochastic setting where we assume that X(k)˜

(0, P·Id_(N)), for k∈

. We denote by X^(s) the following soft estimator for the vector X (0):

$\begin{matrix}{X^{s} = {{\sum\limits_{l = L_{F}}^{R}\;{{F(l)}*{X(l)}}} - {\sum\limits_{l = L_{B}}^{- 1}\;{{B(l)}*{{X(l)}.}}}}} & 1.21\end{matrix}$

We denote by Err=Err (0) the soft error term:Err=X ^(s) −X(0).  1.22

We consider the cost function:U(F,B)=

∥Err∥² =

∥X ^(s) −X(0)∥²,where the expectation is taken over the probability distribution of theinput stream X and the additive white Gaussian noise

. The optimal filters F^(opt), B^(opt) are defined as:

${\left( {F^{opt},B^{opt}} \right) = {\arg\;{\underset{({F,B})}{\;\min}{U\left( {F,B} \right)}}}},$therefore they satisfy the following system or linear equations:∇_(F(l)) U(F ^(opt) ,B ^(opt))=0,l=L _(F) , . . . ,R _(F).∇_(B(l)) U(F ^(opt) ,B ^(opt))=0,l=L _(B), . . . ,−1,  1.23where the gradients are given by:∇_(F(l)) U=

[Err*Y(l)^(★) ],l=L _(F) , . . . ,R _(F),∇_(B(l)) U=−

[Err*X(l)^(★) ],l=L _(B), . . . ,−1.  1.24

First we write explicitly the first system ∇_(F(d))U(F^(opt),B^(opt))=0. Expanding the term

[Err*Y(l)★], we obtain:

${\left\lbrack {{Err}*{Y(l)}^{\bigstar}} \right\rbrack} = {{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{{F\left( l^{\prime} \right)}*{\left\lbrack {{Y\left( l^{\prime} \right)}*{Y(l)}^{\bigstar}} \right\rbrack}}} - {\sum\limits_{l^{\prime} = L_{B}}^{- 1}\;{{B\left( l^{\prime} \right)}*}}}$

Direct computation reveals that:

[X(0) * Y(l)^(★)] = NP ⋅ C(l)^(★), [X(l^(′)) * Y(l)^(★)] = NP ⋅ C(l − l^(′)), [Y(l^(′)) * Y(l)^(★)] = NP ⋅ R₁(l, l^(′)), where:${R_{1}\left( {l,l^{\prime}} \right)} = \left\{ {\begin{matrix}{\sum\limits_{\underset{{s - r} = {l - l^{\prime}}}{s,{r = L_{C}}}}^{R_{C}}\;{{C(r)}*{C(s)}^{\bigstar}}} & {l \neq l^{\prime}} \\{{\sum\limits_{s = L_{C}^{\prime}}^{R_{C}}\;{{C(s)}*{C(s)}^{\bigstar}}} + {\frac{1}{SNR} \cdot \delta_{w = 0}}} & {l = l^{\prime}}\end{matrix}.} \right.$

Thus, the first system of equations amounts to:

$\begin{matrix}{{{{{{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{{R_{1}\left( {l,l^{\prime}} \right)}*{F^{opt}\left( l^{\prime} \right)}}} - {\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}{{C\left( {l - l^{\prime}} \right)}*{B^{opt}\left( l^{\prime} \right)}}}} = {C(l)}^{\bigstar}}, l}\quad} = {\quad{{L_{F}\mspace{14mu}\ldots}\mspace{14mu},{R_{F} \cdot}}}} & 1.25\end{matrix}$

Next we write explicitly the system ∇_(B(l))U(F^(opt), B^(opt))=0.Expanding the term

[Err*X(l)★], we obtain:

${{??}\left\lbrack {{Err}*{X(l)}^{\bigstar}} \right\rbrack} = {{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{{F\left( l^{\prime} \right)}*{{??}\left\lbrack {{Y\left( l^{\prime} \right)}*{X\left( l^{\prime} \right)}^{\bigstar}} \right\rbrack}\underset{l^{\prime} = L_{B}}{\overset{- 1}{- \sum}}{B\left( l^{\prime} \right)}*{{??}\left\lbrack {{X\left( l^{\prime} \right)}*{X(l)}^{\bigstar}} \right\rbrack}}} - {{??}{\left\lceil {{X(0)}*{X(l)}^{\bigstar}} \right\rceil.}}}$

Direct computation reveals that:

[Y(l′)*X(l)^(★) ]=NP·C(l′−l),

[X(l′)*X(l)^(★) ]=NP·δ _(l=l′)·δ_(w=0),

[X(0)*X(l)^(★)]=0,

Thus, the second system of equations amounts to:

$\begin{matrix}{{{{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{{C\left( {l^{\prime} - l} \right)}*{F^{opt}\left( l^{\prime} \right)}}} - {B^{opt}(l)}} = 0},{l = L_{B}},\ldots\mspace{14mu},{- 1.}} & 1.26\end{matrix}$

Using Equation (1.26), the optimal feedback filter taps may be expressedin terms of the channel taps and the optimal forward filter taps as:

$\begin{matrix}{\;\begin{matrix}{{{B^{opt}(l)} = {\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}{{C\left( {l^{\prime} - l} \right)}*{F^{opt}\left( l^{\prime} \right)}}}},{l = L_{B}},\ldots\mspace{14mu},{- 1}}\end{matrix}} & 1.27\end{matrix}$

Substituting the right hand side of (1.27) in (1.25) enables the optimalforward filter taps to be determined by finding the solution of thefollowing linear system:

$\begin{matrix}{{{{{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{{R_{1}\left( {l,l^{\prime}} \right)}*{F^{opt}\left( l^{\prime} \right)}}} - {\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}{{R_{2}\left( {l,l^{\prime}} \right)}*{F^{opt}\left( l^{\prime} \right)}}}} = {C(l)}^{\bigstar}},{l = {L_{F\mspace{14mu}}\ldots}}\mspace{14mu},R_{F},{{where}\text{:}}}{{R_{2}\left( {l,l^{\prime}} \right)} = {\underset{{s - r} = {l - l^{\prime}}}{\sum\limits_{s = {l + 1}}^{l - L_{B}}\sum\limits_{\;{r = {l^{\prime} + 1}}}^{l^{\prime} - L_{B}}}{C(r)}*{{C(s)}^{\bigstar}.}}}} & 1.28\end{matrix}$

As a final note, we denote R(l,l′)=R₁(l,l′)−R₂(l,l′) and write thesystem (1.28) in the following form:

$\begin{matrix}\begin{matrix}{{{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{{R\left( {l,l^{\prime}} \right)}*{F^{opt}\left( l^{\prime} \right)}}} = {C(l)}^{\bigstar}},{l = L_{F}},\ldots\mspace{14mu},R_{F}}\end{matrix} & 1.29\end{matrix}$

System (1.29) can be reduced to N systems of n_(F)=R_(F)−L_(F)=1 scalarequations as follows. Applying a DFT to both sides of (1.15) we obtain:

$\begin{matrix}{{{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\; \cdot} = =},{l = L_{F}},\ldots\mspace{14mu},{R_{F}.}} & 1.30\end{matrix}$

Where (

) stands for the DFT of the corresponding function and · stands forpointwise multiplication of functions in

(R_(d)), since the DFT interchanges convolution with pointwisemultiplication and ★ with complex conjugation. It is now observes thateach function valued equation in (1.30) decouples into n_(F) scalarvalued equations by evaluating both sides on each element of the ringR_(d). Explicitly, numbered the elements in R_(d) as 0, 1, 2, . . . ,N−1 results in the following scalar valued system of equations:

$\begin{matrix}{{{\sum\limits_{l^{\prime} = L_{F}}^{R_{F}}\;{\cdot (i)}} = {(i)}},{l = L_{F}},\ldots\mspace{14mu},R_{F},} & 1.31\end{matrix}$

for every i=0, . . . , N−1. In more concrete matrix form (1.31) lookslike:

( R ⁡ ( F ) ⁢ ( i ) . . R ⁡ ( F ) ⁢ ( i ) . . . . . . . . R ⁡ ( F ) ⁢ ( i ) .. R ⁡ ( F ) ⁢ ( i ) ) ⁢ ( ⁢ ( i ) . . ⁢ ( i ) ) = ( ⁢ ( i ) . . ⁢ ( i ) ) ,1.32

for every i=0, . . . , N−1.

Channel Acquisition

An exemplary channel acquisition component of the OTFS modulation schemewill now be described. To this end, we number the elements of R_(d) by0, 1, . . . , N−1. For the channel acquisition, a rectangular strip [0,R_(C)−2L_(C)]×[0, N] is devoted in the time frequency plane. The valueof the input stream X at this strip is specified to be:

${X\left( {\tau,w} \right)} = \left\{ {\begin{matrix}N & {\tau = {{{- L_{c}}\mspace{14mu}{and}\mspace{20mu} w} = 0}} \\0 & {otherwise}\end{matrix}.} \right.$The complement of this stream will generally be devoted to data.

Gradient Correction

As mentioned previously, the forward and feedback taps of the decisionfeedback equalizer depend on the index k and change slowly as k varies.We proceed to describe herein an exemplary tracking mechanism based ongradient correction with respect to an appropriate quadratic costfunction. We denote by Err (k) the soft error term at the k step:Err(k)=(k)−X ^(s)(k)−X ^(h)(k)∈

(R _(d)),  1.4where theoretically this error should be taken with respect to the truedata vector X (k) (true decisions); however, in an exemplary embodimentthe error is taken with respect to the hard estimator X^(h) (k) (harddecisions) as specified in equation (1.4). We define the following costfunction U, taking as arguments the forward and feedback filter taps:U(F,B)=∥Err(k)∥² =∥X ^(s)(k)−X ^(h)(k)∥²,  1.5where ∥-∥ is the norm associated with the standard Hermitian innerproduct (-,-) on

(R_(d)). Note that, in fact, the cost function depends on the index k,however, for the sake of brevity we omit this index from the notation.Next, we compute the gradients ∇_(F(l))U, l=L_(F), . . . , R_(F) and∇_(B(l))U, l=L_(B, −)1 with respect to the Euclidean inner product 2 Re

-,-

on

(R_(d)). (considered as a real vector space). The formulas for thegradients are:∇_(F(l))=∇_(F(l)) U=Err(k)*Y(k+l)^(★) ,l=L _(F) , . . . ,R _(F),∇_(B(l))=∇_(B(l)) U=−Err(k)*X ^(h))(k+l)^(★) ,l=L _(B), . . . ,−1,  1.6where ★ stands for the star operation on the convolution algebra

(R_(d)), given by f★ (i)=f(−i) for every f∈

(R_(d)) and i∈R_(d). In other words, the star operation of a function isobtained by inverting the coordinate inside R_(d) followed by complexconjugation. We note that the star operation is related by DFT tocomplex conjugation, that is DFT (f★)=DFT (f), for every f∈

(R_(d)).

The correction of the taps at the k step is obtained by adding a smallincrement in the (inverse) gradient direction, that is:F _(k+1)(l)=F _(k)(l)−μ·∇_(F(l)) ,l=L _(F) , . . . ,R _(F),B _(k+1)(l)=B _(k)(l)−μ·∇_(F(l)) ,l=L _(B), . . . ,−1,for an appropriately chosen positive real number μ«1. The optimal valueμ_(opt) of the small parameter μ is given by:

$\mu_{opt} = {\arg{\underset{\mu}{\;\min}{{U\left( {{F + {\mu \cdot \nabla_{F}}},{B + {\mu \cdot \nabla_{B}}}} \right)}.}}}$

A formal development of the quadratic expression U(F+μ·∇_(F), B+μ·∇_(B))in the parameter μ reveals that:

$\begin{matrix}{{U\left( {{F + {\mu \cdot \nabla_{F}}},{B + {\mu \cdot \nabla_{B}}}} \right)} = {{U\left( {F,B} \right)} + {\mu\left( {{2{Re}\left\langle {F,F} \right\rangle} + {2{Re}\left\langle {B,B} \right\rangle}} \right)} +}} \\{{\mu^{2}{{Hess}\left( {\nabla_{F}{,\nabla_{B}}} \right)}},}\end{matrix}$where Hess (∇_(F), ∇_(B)) stands for:

${{Hess}\left( {\nabla_{F}{,\nabla_{B}}} \right)} = {{{\sum\limits_{l = L_{F}}^{R_{F}}{\nabla_{F{(l)}}{*{Y\left( {k + l} \right)}}}} - {\sum\limits_{l - L_{B}}^{- 1}{\nabla_{B{(l)}}{*{X^{h}\left( {k + l} \right)}}}}}}^{2}$and

F,F

and

B,B

stand for:

$\begin{matrix}{{\left\langle {F,F} \right\rangle = {\sum\limits_{l = L_{F}}^{R_{F}}\left\langle {{F(l)},{F(l)}} \right\rangle}},} \\{\left\langle {B,B} \right\rangle = {\sum\limits_{l = L_{B}}^{- 1}{\left\langle {{B(l)},{B(l)}} \right\rangle.}}}\end{matrix}$

If we denote b=2 Re

F,F

+2 Re

B,B

and a=Hess (∇_(F), ∇_(B)) then the standard formula for the minimum of aparabola imply that μ_(opt) is given by:

$\mu_{opt} = {\frac{- b}{2a}.}$

FIG. 33 shows a time-frequency graph providing an illustration of thevarious echo (time shifts) and frequency shifts which a signal mayencounter during transmission through a channel; that is, FIG. 33illustrates the impulse response of the channel. If the channel lackedany echo (time shift) or frequency shifts, signal spike 3400—which isrepresentative of original signal as transformed by the channel—wouldinstead show up as a single spike at a defined time and frequency.However due to various echoes and frequency shifts, the original signalis instead spread over both time 3302 and frequency 3304 in the mannerillustrated by spike 3400. It is thus desired to compensate or otherwiseaddress these effects, either before further processing at the receiver3204 or later after the receiver has taken the processing to the D^(R)stage 3206. Alternatively, the original signal may be pre-equalized 3208prior to transmission using a related process.

FIG. 34 illustratively represents a time-frequency map of tap valuesproduced by the feed forward (FF) portion of the adaptive decisionfeedback equalizer of FIG. 32C when correcting for the time andfrequency distortions introduced by the channel impulse response shownin FIG. 33. The FF portion 3210 of the equalizer works to shift the echoor frequency shifted signals to once again coincide with the main signal(the un-reflected and non-shifted signal), and thus enhances theintensity of the received signal while diminishing the intensity of theecho or frequency shifted signals.

FIG. 35 illustratively represents a time-frequency map of tap valuesproduced by the feedback (FB) portion 3212 of the adaptive decisionfeedback equalizer of FIG. 32C when correcting for the time andfrequency distortions introduced by the channel impulse response shownin FIG. 33. After the feedforward (FF) portion 3210 of the equalizer hasacted to substantially offset the echo and frequency shifted signals,there will still be some residual echo and frequency signals remaining.The feedback (FB) portion 3212 acts to cancel out those trace remainingecho signals, essentially acting like an adaptive canceller for thisportion of the system.

The quantizer portion of the adaptive decision feedback equalizer 3214then acts to “round” the resulting signal to the nearest quantized valueso that, for example, the symbol “1” after transmission, once moreappears on the receiving end as “1” rather than “0.999”.

As previously discussed, an alternative mathematical discussion of theequalization method, particularly suitable for step 802B, is describedin provisional application 61/615,884, the contents of which areincorporated herein by reference.

Data Interleaving

Attention is now directed to FIGS. 36A and 36B, to which reference willbe made in further elaborating upon the use of interleaving within anOTFS system. In particular, FIGS. 36A and 36B show that it may be usefulto transmit various different time blocks in an interleaved scheme wherethe time needed to transmit all N blocks may vary between different datamatrices D, and wherein the interleaving scheme is such as to take thelatency, that is the time needed to transmit all N blocks, into accountaccording to various optimization schemes. By choosing groups oflatencies properly, one can prevent delays to one user or another. Forexample, FIG. 36A shows a first latency timeline 3600 depictingtransmission times for five users a, b, c, d and e. Constellation 3605shows a hierarchical diagram showing two groups including a first groupcomprising users a and b, each with a latency of four, and a secondgroup comprising users c, d and e, each with a latency of six. Thismeans that users a and b will transmit, or receive data every four timeslots, while users c, d and e will transmit or receive data every sixtime slots. Time track 3610 shows the resulting order oftransmission/receiving for each user, while latency indicators 3615,3620, 3625, 3630 and 3635 show the resulting latency spacing for usersa, b, c, d and e, respectively.

FIG. 36B shows a second latency timeline 3650 showing the transmissiontimes for four users a, b, c and d. Constellation 3655 shows ahierarchical diagram depicting three groups including a first groupcomprising user a with a latency of two, a second group comprising userb with a latency of four, and a third group comprising users c and d,each with a latency of eight. This means that user a will transmit orreceive data every two time slots, user b will transmit, or receive dataevery four time slots, while users c and d will transmit or receive dataevery eight time slots. Time track 3660 shows the resulting order oftransmission/receiving for each user, while latency indicators 3665,3670, 3675 and 3680 show the resulting latency spacing for users a, b, cand d, respectively. Different latencies can be chosen for differentusers depending on what type of service the user is seeking. Forexample, a voice connection may be provided a latency of two, while afile or video download might be provided a latency of eight. Latency maybe chosen for other reasons.

Full Duplex Transceiver

FIG. 37 shows an example of a full duplex OTFS transceiver 3700 capableof enabling data to be transmitted and received simultaneously in thesame frequency band. The OTFS transceiver 3700 is configured with anecho cancellation module 3705 that implements echo cancelation in thetime and frequency domain. This enables estimation of two-dimensionalreflections of the transmitted signal; that is, estimation of frequencyshifts and time shifts. As shown, a first OTFS encoder 3710-1 performsOTFS encoding with a first matrix [U1], a permutation operation, asecond matrix multiplication of a Basis matrix [U2] and sin/costransmission of the elements of the resulting transformed data matrix.The transformed data matrix is transmitted a column at a time in a onedimensional data stream and up-converted to an RF frequency with RF upconverter 3715-1, power amplified with transmit power amplifier 3720-1and passed to an antenna 3740 via a circulator 3722.

In the embodiment of FIG. 37 the antenna also receives a second datastream from another transmitter. However, the second data stream alsoincludes reflections of the first signal transmitted by the OTFStransmitter 3700. The circulator 3722 routes the received second signalto a subtractor 3724 that subtracts an estimate of the reflected signalsthat is created by the echo canceller 3705. A second OTFS encoder3710-2, a second RF up converter 3715-2 and an echo canceller poweramplifier 3720-2 create the estimated echo that is subtracted from thereceived second signal.

An RF down converter 3725 demodulates the second received signal andpasses the demodulated received second data stream Dr to a first OTFSdecoder 3730-1 and a second OTFS decoder 3730-2. The first OTFS decoder3730-2 decodes the received second signal using the base t matrix thatwas used to transmit the first data stream. The second OTFS decoder3730-2 decodes the echo-canceled data stream using a base r matrix thatthe other transmitter used to encode the second data stream. The outputof the first OTFS decoder 3730-1 is fed back as a residual error signalto the echo canceller 3705 in order to tune the two dimensional estimateof the reflected echoes channel. The output of the second OTFS decoder3730-2 is an estimate of the second data stream from the othertransmitter. The capability to obtain an estimate of the echo channel inboth frequency and time is a significant advantage of the OTFStechnique, and facilitates full-duplex communication over a commonfrequency band in a manner not believed to be possible using prior artmethods.

Iterative Signal Separation

FIG. 38 shows an example of an OTFS receiver 3800 that providesiterative signal separation in accordance with the disclosure. The OTFSreceiver 3800 receives a first data matrix D₁ from a first transmitterthat uses a first basis matrix. The OTFS receiver 3800 also receives asecond data stream D₂ from a second transmitter in the same frequencyband where the second data stream D₂ was encoded using a second basismatrix different from the first basis matrix. A first OTFS decoder3810-1 decodes the first data matrix D₁ to create a one dimensional datastream Y₁ while a second OTFS decoder decodes the second data matrix D₂to form a second one dimensional data stream Y₂.

The OTFS receiver 3800 includes a pair of feed-forward and feedbackequalizers comprising first and second feed forward equalizers 3820-1and 3820-2, first and second feedback equalizers 3835-1 and 3835-2, andfirst and second slicers 3825-1 and 3825-2. First and second subtractors3830-1 and 3830-2 calculate first and second residual error signals3840-1 and 3840-2 that are used by respective ones of the feed forwardequalizers 3820 and the feedback equalizers 3835 in order to optimizetwo dimensional time/frequency shift channel models.

A pair of cross talk cancellers 3845-1 and 3845-2 also use the residualerror signals 3840-1 and 3840-2, respectively, in order to optimizeestimates of the first received data signal and the second received datasignal in order to subtract each signal at subtractors 3815-1 and3815-2. In this way, the cross talk from one data signal to the other isminimized. As with the full duplex OTFS transceiver 3700 of FIG. 37, theOTFS receiver 3800 can model two dimensional time/frequency channels andis believed to represent a significant advance over receivers employingconventional one dimensional (i.e., time only) channel modelingapproaches.

Attention is now directed to FIG. 40, which is a block diagram of atime-frequency-space decision feedback equalizer 4000 that may beemployed to facilitate signal separation in a multi-antenna OTFS system.As shown in FIG. 40, received signal information (R) represented by aset of M time-frequency planes 4004 is received at input port 4010 ofthe equalizer 4000. Each of the M time-frequency planes 4004 representsthe information collected from N transmit antenna instances (M>N) by oneof M antenna instances associated with an OTFS receiver. The N transmitantenna instances, which may or may not be co-located, will generally beassociated with an OTFS transmitter remote from the OTFS receiverassociated with the M receive antenna instances. Each of the N transmitantenna instances and M receive antenna instances may, for example,comprise a single physical antenna which is either co-located or notco-located with the other antenna instances. Alternatively, one or moreof the N transmit antenna instances and M receive antenna instances maycorrespond to an antenna instance obtained through polarizationtechniques.

In the embodiment of FIG. 40, the time-frequency-space decision feedbackequalizer 4000 includes a time-frequency-space feedforward FIR filter4020 and a time-frequency-space feedback FIR filter 4030. The equalizer4000 produces an equalized data stream at least conceptually arranged inset of N time-frequency planes (M>N) wherein, again, N corresponds tothe number of antenna instances transmitting information to the Mantenna instances of the OTFS receiver associated with the equalizer4000.

Turning now to FIG. 41, a block diagram is provided of atime-frequency-space decision feedforward FIR filter 4100 which may beutilized to implement the time-frequency-space feedforward FIR filter4020. As shown, the filter 4100 processes received signal information(R) carried on a set of M time-frequency planes 4104 provided by acorresponding set of M receive antennas. The filter 4100 produces afiltered data stream at least conceptually arranged in set of Ntime-frequency planes 4150 (M>N), where, again, N corresponds to thenumber of antenna instances transmitting information to the M antennainstances of the OTFS receiver associated with the equalizer 4000.

Referring to FIG. 42, a block diagram is provided of atime-frequency-space decision feedback FIR filter 4200 which may beutilized to implement the time-frequency-space feedback FIR filter 4030.As shown, the filter 4200 processes received signal information (R)carried on a set of M time-frequency planes 4204 which may, for example,correspond to the set of M time-frequency planes provided by acorresponding set of M receive antennas. The filter 4200 produces afiltered data stream at least conceptually arranged in a set of Ntime-frequency planes 4250 (M>N).

The time-frequency-space decision feedback equalizer 4000 advantageouslyenables the separation of signals within an OTFS communication system ina manner that substantially maximizes utilization of the availablebandwidth. Such signal separation is useful in several contexts withinan OTFS communication system. These include separation, at a receiverfed by multiple co-located or non-co-located antennas, of signalstransmitted by a set of co-located or non-co-located antennas of atransmitter. In addition, the time-frequency-space decision feedbackequalizer 4000 enables the separation, from signal energy received froma remote transmitter, of echoes received by a receive antenna inresponse to transmissions from a nearby transmit antenna. This echocancellation may occur even when the transmit and receive signal energyis within the same frequency band, since the two-dimensionalchannel-modeling techniques described herein enable accurate andstationary representations of the both the echo channel and the channelassociated with the remote transmitter. Moreover, as is discussed belowthe signal separation capability of the disclosed time-frequency-spacedecision feedback equalizer enables deployment of OTFS transceivers in amesh configuration in which neighboring OTFS transceivers may engage infull duplex communication in the same frequency band with other suchtransceivers in a manner transparent to one another.

Again with reference to FIG. 40, operation of an exemplary OTFS systemmay be characterized as the transmission, from each antenna instanceassociated with a transmitter, of a time-frequency plane representing atwo-dimensional information array being sent. Each such antennainstance, whether co-located or non-co-located, may simultaneouslytransmit two-dimensional information planes, each independent of theother. The information in each of these information planes may beshifted in time and frequency using the same basis functions. Duringtransmission from each of N transmit antenna instances to each of Mreceive antenna instances, the information within each transmitted planeis differently affected by the different two-dimensional channelslinking one of the N transmit antenna instances to each of the M receiveantenna instances.

At each of the M antenna instances associated with an OTFS receiver,each entry within the two-dimensional array of received signal energybeing collected will typically include a contribution from each of the Ntransmit antenna instances involved in transmitting such signal energy.That is, each of the M receive antenna instances collects a mixture ofthe two-dimensional, time-frequency planes of information separatelysent by each of the N transmit antenna instances. Thus, the problem tobe solved by the equalizer 4000 may be somewhat simplisticallycharacterized as inversion of the N×M “coupling matrix” representativeof the various communication channels between the N OTFS transmitantenna instances and the M OTFS receive antenna instances.

In one embodiment each of the N transmit antenna instances sends a pilotsignal which may be differentiated from the pilot signals transmitted bythe other N−1 antenna instances by its position in the time-frequencyplane. These pilot signals enable the OTFS receiver to separatelymeasure each channel and the coupling between each antenna instance.Using this information the receiver essentially initializes the filterspresent within the equalizer 4000 such that convergence can be achievedmore rapidly. In one embodiment an adaptive process is utilized torefine the inverted channel or filter used in separating the receivedsignal energy into different time-frequency-space planes. Thus, thecoupling channel between each transmit and receive antenna instance maybe measured, the representation of the measured channel inverted, andthat inverted channel representation used to separate the receivedsignal energy into separate and distinct time-frequency planes ofinformation.

As noted above, the channel models associated with known conventionalcommunication systems, such as OFDM-based systems, are one-dimensionalin nature. As such, these models are incapable of accurately taking intoconsideration all of the two-dimensional (i.e., time-based andfrequency-based) characteristics of the channel, and are limited toproviding an estimate of only one such characteristic. Moreover, suchone-dimensional channel models change rapidly relative to the time scaleof modern communication systems, and thus inversion of the applicablechannel representation becomes very difficult, if possible at all.

The stationary two-dimensional time-frequency channel models describedherein also enable OFTS systems to effectively implementcross-polarization cancellation. Consider the case in which a transmitantenna instance associated with an OFTS transceiver is configured forhorizontally-polarized transmission and a nearby receive antenna of theOFTS transceiver is configured to receive vertically-polarized signalenergy. Unfortunately, reflectors proximate either the transmit orreceive antenna may reflect and cross-polarize some of the transmittedhorizontally-polarized energy from the transmit antenna, some of whichmay be directed to the receive antenna as a vertically-polarizedreflection. It is believed that a two-dimensional channel model of thetype disclosed herein is needed in order to decouple and cancel thiscross-polarized reflection from the energy otherwise intended for thereceive antenna.

Similarly, full duplex communication carried out on the same channelrequires echo cancellation sufficiently robust to substantially removethe influence of a transmitter on a nearby receiver. Again, such echocancellation is believed to require, particularly in the case of movingreflectors, an accurate two-dimensional representation of at least theecho channel in order to permit the representation to be appropriatelyinverted.

OTFS Transceiver Using Spreading Kernel

As discussed above, embodiments of the OTFS method may involvegenerating a two-dimensional matrix by spreading a two-dimensional inputdata matrix. In addition, time/frequency tiling may be utilized intransport of the two-dimensional matrix across a channel. In thisapproach each matrix column may be tiled as a function of time; that is,each column element occupies a short symbol time slice utilizing thefull available transmission bandwidth, with time gaps optionallyinterposed between subsequent columns. Alternatively, the matrix columnsmay be tiled and transported as a function of frequency; that is, eachelement of the column occupies a frequency bin for a longer period oftime, with time gaps optionally interposed between subsequent columns.

In other embodiments a spreading kernel may be used to effect spreadingof the input data matrix. In this case two-dimensional spreading may beachieved through, for example, a two-dimensional cyclic convolution witha spreading kernel, a convolution implemented using a two-dimensionalFFT, multiplication with the two-dimensional DFT of the spreadingkernel, followed by a two-dimensional inverse Fourier transform. A widevariety of spreading kernels may be utilized; however, thetwo-dimensional DFT of the selected kernel should lack any zeroes so asto avoid division by zero during the dispreading process. Moreover,spreading may also be achieved using alternate methods of convolutions,transforms and permutations. Masking (i.e., element by elementmultiplication) may also be utilized as long as each operation ininvertible.

Attention is now directed to FIGS. 44A and 44B, which provide blockdiagram representations of embodiments of a first OTFS transceiver 4400and a second OTFS transceiver 4450 configured to utilize a spreadingkernel. Reference will be made to the first OTFS transceiver 4400 ofFIG. 44A in describing principles of OTFS communication using aspreading kernel. The second OTFS transceiver 4450 is substantiallysimilar in principle to the first OTFS transceiver 4400 but ischaracterized by an architecture believed to enable more efficientsignal processing.

As shown in FIG. 44A, a transmitter 4404 of the first OTFS transceiver4400 includes two-dimensional spreading block 4408, an FFT block 4410and first and second time-frequency tiling elements 4412, 4414. Thefirst and second time-frequency tiling elements 4412, 4414 areconfigured to effect time-frequency tiling of the two-dimensionallyspread input data and may, for example, be implemented using one or morefilter banks. The two-dimensional spreading block 4408 and FFT block4410 cooperatively effect spreading of the two-dimensional input data byperforming a series of operations using, for example, a spreading kernelselected from a wide family of unitary matrices. In one embodiment thisseries of operations includes two-dimensional cyclic convolution withthe spreading kernel, convolution implemented using a two-dimensionalFFT, multiplication using two-dimensional Discrete Fourier Transform ofthe spreading kernel, and a two-dimensional inverse Fourier transform.This results in cyclically shifting the kernel matrix “up” along thecolumn direction by an amount corresponding to an information index(yielding a time shift) and multiplying by a diagonal tone whosefrequency is set by the information index. All resulting transformedmatrices are then summed together in order to generate thetwo-dimensional spread matrix, each element of which is carried using atransformed Kernel (basis matrix).

A receiver 4420 of the first OTFS transceiver 4400 includes first andsecond inverse time-frequency tiling elements 4424, 4426 configured toeffect an inverse of the tiling operation performed by thetime-frequency tiling elements 4412 and 4414. A two-dimensional IFFTblock 4428 and despreading block 4430 are configured to perform theinverse of the spreading operation performed by the two-dimensionalspreading block 4408 and the FFT block 4410. The received data is thenconverted using an FFT block 4434 prior to being equalized by atime-frequency-space decision feedforward/feedback analyzer block 4438.The equalized data is then converted using an IFFT block 4440.

Turning now to FIG. 44B, a transmitter 4454 of the second OTFStransceiver 4450 includes a two-dimensional spreading arrangementcomprised of an FFT block 4458 and a multiplier 4460 addressed by aFourier mask. Within the transmitter 4454, each information element isrepresented as a cyclic shift of a kernel matrix in both a horizontal(row) and vertical (column) direction corresponding to the applicableinformation element index (row and column position in the inputtwo-dimensional information array). In the implementation of FIG. 44B,the spreading kernel is selected such that its two-dimensional DFT iscomprised entirely of non-zero elements (thus enabling the resultingmatrix to be inverted without forming singularities). The resultingmatrix goes through a DFT transformation of the rows to represent thetwo-dimensional spread information element. All resulting transformedmatrices are then summed together in order to generate the resultingtwo-dimensional spread information matrix.

As shown in FIG. 44B, an arrangement of time-frequency tiling elements4462, 4464 and 4466 are configured to effect time-frequency tiling ofthe two-dimensionally spread input data output by the multiplier 4460.The time-frequency tiling elements 4464 and 4466 may, for example, beimplemented using one or more filter banks.

A receiver 4470 of the second OTFS transceiver 4450 includes a serialarrangement of inverse time-frequency tiling elements 4474, 4476, 4478configured to effect an inverse of the tiling operation performed by thetime-frequency tiling elements 4462, 4464 and 4466. A multiplier 4480 isconfigured to multiply the output produced by the inverse time-frequencytiling elements 4474, 4476 and 4478 by an inverse mask. Next, an IFFTblock 4482 converts the output of the multiplier 4480 and provides theresult to a time-frequency-space decision feedforward/feedback analyzerblock 4488. The equalized data is then converted by an IFFT block 4492.

Mesh Networking

Attention is now directed to FIGS. 50-52, which illustratively representmesh network implementations of OTFS communication systems. The OTFSmesh networks depicted in FIGS. 50-52 advantageously leverage thetime-frequency-space equalization and echo cancellation techniquesdescribed herein to enable OTFS mesh nodes to engage in full duplexcommunication with other such nodes on the same communication channel,whether or not such communication channel is also used by neighboringOTFS mesh nodes.

Referring to FIG. 50, there is shown an OTFS mesh network 5000 withinthe context of a cellular communication system comprised of cell sites5004 and associated cell coverage areas 5008. As may be appreciated fromFIG. 50, significant gaps may exist between the coverage areas 5008.

The mesh network 5000 comprises a plurality of OTFS wireless mesh nodes5020 operative to provide wireless communication coverage to fixed ormobile devices within areas of high demand which are generally outsideof the range of the coverage areas 5008. For the reasons discussedabove, each OTFS wireless mesh node 5020 may be configured for fullduplex wireless communication with other such mesh nodes 5020 over thesame frequency band. This full duplex wireless communication over thesame frequency band is represented in FIG. 50 by wireless communicationlinks 5030. In the embodiment of FIG. 50 each of the wireless mesh links5030 operates over an identical frequency range.

Turning now to FIG. 51, there is shown an OTFS mesh network 5100organized around a set of wired network gateways 5110. The mesh network5100 comprises a plurality of OTFS wireless mesh nodes 5120 operative toprovide wireless communication to fixed or mobile devices within areasproximate each of the nodes 5120. Each OTFS wireless mesh node 5120 maybe configured for full duplex wireless communication with other suchmesh nodes 5120 over the same frequency band. This full duplex wirelesscommunication over the same frequency band is represented in FIG. 51 bywireless mesh links 5130. In the embodiment of FIG. 51, the wirelessmesh nodes 5120 are self-organizing in the sense that the nodes 5120 areconfigured to discover each other and to determine all possible pathsover links 5130 to each wired network gateway 5110. Accordingly, networkrouting techniques may be employed to route packetized informationbetween and among the mesh nodes 5120 and the wired network gateways5110 in both directions over the wireless mesh links 5130.

FIG. 52 shows an OTFS mesh network system 5200 comprised of asingle-channel wireless mesh network 5204 including plurality of meshelements. In one embodiment certain of the mesh elements of mesh network5204 preferably include an OTFS wireless mesh router 5210 and a trafficaggregation device 5220 (e.g., and LTE node or Wi-Fi access point)serving end user devices 5250 within a respective coverage area 5254.Each OTFS wireless mesh router 5210 may be configured for full duplexwireless communication with other such mesh nodes 5210 over the samefrequency band. In the embodiment of FIG. 52, the wireless mesh nodes5210 are self-organizing in the sense that the nodes 5210 are configuredto discover each other and to determine all possible paths over OTFSwireless links 5230 to each wired network gateway 5240. Accordingly,network routing techniques may be employed to route packetizedinformation between and among the mesh nodes 5120 and a wired network5244—via the wired network gateways 5110—in both directions over thewireless mesh links 5130. As shown, the wired network 5244 may provide aconduit to a wide area network through which information packets arerouted between the mesh network 5204 and a core network 5260 of a mobilenetwork operator.

In one embodiment mesh spatial gain may be achieved by using neighboringmesh nodes 5120 to support the simultaneous parallel transmission ofstreams of information using an identical frequency band over a singlepoint to point link. This approach may improve signal transmission gainby using neighboring nodes 5120 to effectively create a distributedtransmit source, thereby achieving gain through spatial signalseparation.

Some embodiments of the systems and methods described herein may includecomputer software and/or computer hardware/software combinationsconfigured to implement one or more processes or functions associatedwith the methods such as those described above and/or in the relatedapplications. These embodiments may be in the form of modulesimplementing functionality in software and/or hardware softwarecombinations. Embodiments may also take the form of a computer storageproduct with a computer-readable medium having computer code thereon forperforming various computer-implemented operations, such as operationsrelated to functionality as describe herein. The media and computer codemay be those specially designed and constructed for the purposes of theclaimed systems and methods, or they may be of the kind well known andavailable to those having skill in the computer software arts, or theymay be a combination of both.

Examples of computer-readable media within the spirit and scope of thisdisclosure include, but are not limited to: magnetic media such as harddisks; optical media such as CD-ROMs, DVDs and holographic devices;magneto-optical media; and hardware devices that are speciallyconfigured to store and execute program code, such as programmablemicrocontrollers, application-specific integrated circuits (“ASICs”),programmable logic devices (“PLDs”) and ROM and RAM devices. Examples ofcomputer code may include machine code, such as produced by a compiler,and files containing higher-level code that are executed by a computerusing an interpreter. Computer code may be comprised of one or moremodules executing a particular process or processes to provide usefulresults, and the modules may communicate with one another via meansknown in the art. For example, some embodiments of systems describedherein may be implemented using assembly language, Java, C, C#, C++, orother programming languages and software development tools as are knownin the art. Other embodiments of the described systems may beimplemented in hardwired circuitry in place of, or in combination with,machine-executable software instructions.

The foregoing description, for purposes of explanation, used specificnomenclature to provide a thorough understanding of the claimed systemsand methods. However, it will be apparent to one skilled in the art thatspecific details are not required in order to practice the systems andmethods described herein. Thus, the foregoing descriptions of specificembodiments of the described systems and methods are presented forpurposes of illustration and description. They are not intended to beexhaustive or to limit the claims to the precise forms disclosed;obviously, many modifications and variations are possible in view of theabove teachings. The embodiments were chosen and described in order tobest explain the principles of the described systems and methods andtheir practical applications, they thereby enable others skilled in theart to best utilize the described systems and methods and variousembodiments with various modifications as are suited to the particularuse contemplated. It is intended that the following claims and theirequivalents define the scope of the systems and methods describedherein.

What is claimed is:
 1. A method of receiving data, comprising:receiving, on one or more carrier waveforms, signals representing aplurality of data elements of an original data frame wherein each of thedata elements are represented by cyclically time shifted and cyclicallyfrequency shifted versions of a known set of waveforms; demodulating thesignals to form a transformed data frame having a first dimension of atleast N elements and a second dimension of at least M elements, where Nand M are integers greater than one; wherein the first dimensioncorresponds to a frequency shift axis and the second dimensioncorresponds to a time shift axis; performing an inverse time-frequencytransformation operation with respect to elements of the transformeddata frame so as to yield a non-transformed matrix; and generating,based upon the non-transformed matrix, a recovered data frame comprisingan estimate of the original data frame.
 2. The method of claim 1 whereinthe inverse time-frequency transformation is performed using a decodingmatrix, the decoding matrix being a Hermitian matrix of a time-frequencytransformation matrix used to transform the original data frame.
 3. Themethod of claim 1 wherein the generating includes performing an inversepermutation operation upon elements of the non-transformed matrix,wherein the inverse-permutation operation effects a shift of theelements of the inverse-transformed matrix along a time shift axis ofthe non-transformed matrix.
 4. The method of claim 1 wherein N is equalto M.
 5. The method of claim 1 further including generating an equalizeddata frame by performing an equalization operation using elements of thetransformed data frame, the equalization operation correcting fordistortion introduced into the signals during propagation of the carrierwaveforms through a channel.
 6. The method of claim 1 further includingdetecting a time-shifted signal component of the signals and afrequency-shifted signal component of the signals and generating, basedupon the detecting, corrective information.
 7. The method of claim 6further including sending the corrective information to a transmitterfrom which the signals were previously transmitted.
 8. The method ofclaim 6 further using the corrective information to equalize elements ofthe transformed data frame.
 9. A data receiver, comprising: a processor;a memory including program code executable by the processor, the programcode including: code for facilitating receiving, on one or more carrierwaveforms, signals representing a plurality of data elements of anoriginal data frame wherein each of the data elements are represented bycyclically time shifted and cyclically frequency shifted versions of aknown set of waveforms; code for demodulating the signals to form atransformed data frame having a first dimension of at least N elementsand a second dimension of at least M elements, where N and M areintegers greater than one; wherein the first dimension corresponds to afrequency shift axis and the second dimension corresponds to a timeshift axis; code for performing an inverse time-frequency transformationoperation with respect to elements of the transformed data frame so asto yield a non-transformed matrix; and code for generating, based uponthe non-transformed matrix, a recovered data frame comprising anestimate of the original data frame.
 10. The data receiver of claim 9wherein the inverse time-frequency transformation is performed using adecoding matrix, the decoding matrix being a Hermitian matrix of atime-frequency transformation matrix used to transform the original dataframe.
 11. The data receiver of claim 9 wherein the code for generatingincludes code for performing an inverse permutation operation uponelements of the non-transformed matrix, wherein the inverse-permutationoperation effects a shift of the elements of the inverse-transformedmatrix along a time shift axis of the non-transformed matrix.
 12. Thedata receiver of claim 9 wherein N is equal to M.